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Page No 311:

Question 1:

Answer:

  (i) A(9, 3) and B(15, 11)
The given points are A(9, 3) and B(15, 11).
Then (x1 = 9, y1 = 3) and (x2 = 15, y2 = 11)
AB = x2-x12+y2-y12      =15-92+11-32      =15-92+11-32      =62+82      =36+64      =100      = 10 units

(ii) A(7, −4) and B(−5, 1)
The given points are A(7, −4) and B(−5, 1).
Then (x1= 7, y1 = −4) and (x2= −5, y2 = 1)
AB = x2-x12+y2-y12       =-5-72+1--42       =-5-72+1+42       =-122+52       =144+25       =169       = 13 units

(iii) A(−6, −4) and B(9, −12)
The given points are A(−6, −4) and B(9, −12).
Then (x1 = −6, y1 = −4) and (x2 = 9, y2 = −12)
AB = x2-x12+y2-y12      =9--62+-12--42      =9+62+-12+42      =152+-82      =225+64      =289      = 17 units

(iv) A(1, −3) and B(4, −6)
The given points are A(1, −3) and B(4, −6).
Then (x1= 1, y1 = −3) and (x2 = 4, y2 = −6)
AB = x2-x12+y2-y12      =4-12+-6--32      =4-12+-6+32      =32+-32      =9+9      =18      =9×2      = 32 units

(v) P(a + b, ab) and Q(ab, a + b)
The given points are P(a + b, a − b) and Q(a − b, a + b).
Then (x1 = a + b, y1 = a − b) and (x2 = a − b, y2 = a + b)
PQ = x2-x12+y2-y12       =a-b-a+b2+a+b-a-b2       =a-b-a-b2+a+b-a+b2       =-2b2+2b2       =4b2+4b2       =8b2       =4×2b2       = 22b units

(vi) P(a sin α, a cos α) and Q(a cos α, −a sin α)
The given points are P(a sin α, a cos α) and Q(a cos α, −a sin α).
Then (x1 = a sin α, y1 = a cos α) and (x2 = a cos α, y2 = −a sin α)
PQ = x2-x12+y2-y12       =a cos α-a sin α2+-a sinα-a cos α2       =a2cos2α +a2 sin2 α - 2a2cos α×sin α+a2 sin2 α+a2cos2α+2a2cos α× sin α       =2a2cos2α +2a2 sin2 α       =2a2cos2α + sin2 α       =2a21                  From the identity cos2α + sin2 α = 1       =2a2       = 2a units

Page No 311:

Question 2:

  (i) A(9, 3) and B(15, 11)
The given points are A(9, 3) and B(15, 11).
Then (x1 = 9, y1 = 3) and (x2 = 15, y2 = 11)
AB = x2-x12+y2-y12      =15-92+11-32      =15-92+11-32      =62+82      =36+64      =100      = 10 units

(ii) A(7, −4) and B(−5, 1)
The given points are A(7, −4) and B(−5, 1).
Then (x1= 7, y1 = −4) and (x2= −5, y2 = 1)
AB = x2-x12+y2-y12       =-5-72+1--42       =-5-72+1+42       =-122+52       =144+25       =169       = 13 units

(iii) A(−6, −4) and B(9, −12)
The given points are A(−6, −4) and B(9, −12).
Then (x1 = −6, y1 = −4) and (x2 = 9, y2 = −12)
AB = x2-x12+y2-y12      =9--62+-12--42      =9+62+-12+42      =152+-82      =225+64      =289      = 17 units

(iv) A(1, −3) and B(4, −6)
The given points are A(1, −3) and B(4, −6).
Then (x1= 1, y1 = −3) and (x2 = 4, y2 = −6)
AB = x2-x12+y2-y12      =4-12+-6--32      =4-12+-6+32      =32+-32      =9+9      =18      =9×2      = 32 units

(v) P(a + b, ab) and Q(ab, a + b)
The given points are P(a + b, a − b) and Q(a − b, a + b).
Then (x1 = a + b, y1 = a − b) and (x2 = a − b, y2 = a + b)
PQ = x2-x12+y2-y12       =a-b-a+b2+a+b-a-b2       =a-b-a-b2+a+b-a+b2       =-2b2+2b2       =4b2+4b2       =8b2       =4×2b2       = 22b units

(vi) P(a sin α, a cos α) and Q(a cos α, −a sin α)
The given points are P(a sin α, a cos α) and Q(a cos α, −a sin α).
Then (x1 = a sin α, y1 = a cos α) and (x2 = a cos α, y2 = −a sin α)
PQ = x2-x12+y2-y12       =a cos α-a sin α2+-a sinα-a cos α2       =a2cos2α +a2 sin2 α - 2a2cos α×sin α+a2 sin2 α+a2cos2α+2a2cos α× sin α       =2a2cos2α +2a2 sin2 α       =2a2cos2α + sin2 α       =2a21                  From the identity cos2α + sin2 α = 1       =2a2       = 2a units

Answer:

(i) A(5, −12)
Let O(0, 0) be the origin.
 OA = 5-02+-12-02      = 52+-122      = 25+144      = 169      =13 units

(ii) B(−5, 5)
Let O(0, 0) be the origin.
OB = -5-02+5-02       = -52+52       = 25+25       = 50       =25×2       =52 units

(iii) C(−4, −6)
Let O(0,0) be the origin.
OC = -4-02+-6-02       = -42+-62       = 16+36       = 52       =4×13       =213 units

Page No 311:

Question 3:

(i) A(5, −12)
Let O(0, 0) be the origin.
 OA = 5-02+-12-02      = 52+-122      = 25+144      = 169      =13 units

(ii) B(−5, 5)
Let O(0, 0) be the origin.
OB = -5-02+5-02       = -52+52       = 25+25       = 50       =25×2       =52 units

(iii) C(−4, −6)
Let O(0,0) be the origin.
OC = -4-02+-6-02       = -42+-62       = 16+36       = 52       =4×13       =213 units

Answer:

Given AB = 5 units
Therefore, (AB)2 = 25 units
5-x2+3--12 = 255-x2+3+12 = 255-x2+42 = 255-x2+16 = 255-x2 = 25-165-x2 = 95-x =± 95-x= ±35-x = 3  or   5-x=-3x=2 or 8
Therefore, x = 2 or 8.

Page No 311:

Question 4:

Given AB = 5 units
Therefore, (AB)2 = 25 units
5-x2+3--12 = 255-x2+3+12 = 255-x2+42 = 255-x2+16 = 255-x2 = 25-165-x2 = 95-x =± 95-x= ±35-x = 3  or   5-x=-3x=2 or 8
Therefore, x = 2 or 8.

Answer:

The given points are A2,-3 and B10,y.
AB=2-102+-3-y2          =-82+-3-y2          =64+9+y2+6y
AB=1064+9+y2+6y=1073+y2+6y=100        Squaring both sidesy2+6y-27=0
y2+9y-3y-27=0yy+9-3y+9=0y+9y-3=0y+9=0 or y-3=0
y=-9 or y=3
Hence, the possible values of y are -9 and 3.

Page No 311:

Question 5:

The given points are A2,-3 and B10,y.
AB=2-102+-3-y2          =-82+-3-y2          =64+9+y2+6y
AB=1064+9+y2+6y=1073+y2+6y=100        Squaring both sidesy2+6y-27=0
y2+9y-3y-27=0yy+9-3y+9=0y+9y-3=0y+9=0 or y-3=0
y=-9 or y=3
Hence, the possible values of y are -9 and 3.

Answer:

The given points are P(x, 4) and Q(9, 10).
PQ=x-92+4-102          =x-92+-62          =x2-18x+81+36           =x2-18x+117
PQ=10x2-18x+117=10x2-18x+117=100        Squaring both sidesx2-18x+17=
x2-17x-x+17=0xx-17-1x-17=0x-17x-1=0x-17=0 or x-1=0
x=17 or x=1
Hence, the values of x are 1 and 17.

Page No 311:

Question 6:

The given points are P(x, 4) and Q(9, 10).
PQ=x-92+4-102          =x-92+-62          =x2-18x+81+36           =x2-18x+117
PQ=10x2-18x+117=10x2-18x+117=100        Squaring both sidesx2-18x+17=
x2-17x-x+17=0xx-17-1x-17=0x-17x-1=0x-17=0 or x-1=0
x=17 or x=1
Hence, the values of x are 1 and 17.

Answer:

As per the question
AB=ACx-82+2+22=x-22+2+22
Squaring both sides, we get
x-82+42=x-22+42x2-16x+64+16=x2+4-4x+1616x-4x=64-4x=6012=5
Now,
AB=x-82+2+22     =5-82+2+22                x=2     =-32+42     =9+16=25=5
Hence, x = 5 and AB = 5 units.

Page No 311:

Question 7:

As per the question
AB=ACx-82+2+22=x-22+2+22
Squaring both sides, we get
x-82+42=x-22+42x2-16x+64+16=x2+4-4x+1616x-4x=64-4x=6012=5
Now,
AB=x-82+2+22     =5-82+2+22                x=2     =-32+42     =9+16=25=5
Hence, x = 5 and AB = 5 units.

Answer:

As per the question
AB=AC0-32+2-p2=0-p2+2-52-32+2-p2=-p2+-32
Squaring both sides, we get
-32+2-p2=-p2+-329+4+p2-4p=p2+94p=4p=1
Now,
AB=0-32+2-p2     =-32+2-12                p=1     =9+1     =10 units
Hence, p = 1 and AB10 units.



Page No 312:

Question 8:

As per the question
AB=AC0-32+2-p2=0-p2+2-52-32+2-p2=-p2+-32
Squaring both sides, we get
-32+2-p2=-p2+-329+4+p2-4p=p2+94p=4p=1
Now,
AB=0-32+2-p2     =-32+2-12                p=1     =9+1     =10 units
Hence, p = 1 and AB10 units.

Answer:

Let the point on the x - axis be (x, 0).
We have A-2,5 and B2,-3AX=BXAX2=BX2    .....1Using distance formula:d=x2-x12+y2-y12For AXAX=x--22+0-52AX2=x+22+-52AX2=x2+4x+29     .....2
And BX=x-22+0--32BX2=x-22+32BX2=x2-4x+13    .....3From 1,2 and 3x2+4x+29=x2-4x+13x=-2
Point on the x - axis is (-2, 0).

Page No 312:

Question 9:

Let the point on the x - axis be (x, 0).
We have A-2,5 and B2,-3AX=BXAX2=BX2    .....1Using distance formula:d=x2-x12+y2-y12For AXAX=x--22+0-52AX2=x+22+-52AX2=x2+4x+29     .....2
And BX=x-22+0--32BX2=x-22+32BX2=x2-4x+13    .....3From 1,2 and 3x2+4x+29=x2-4x+13x=-2
Point on the x - axis is (-2, 0).

Answer:

Let P (x, 0) be the point on the x-axis. Then as per the question, we have
AP=10x-112+0+82=10x-112+82=100                     Squaring both sidesx-112=100-64=36
x-11=±6x=11±6x=11-6, 11+6x=5, 17
Hence, the points on the x-axis are (5, 0) and (17, 0).

Page No 312:

Question 10:

Let P (x, 0) be the point on the x-axis. Then as per the question, we have
AP=10x-112+0+82=10x-112+82=100                     Squaring both sidesx-112=100-64=36
x-11=±6x=11±6x=11-6, 11+6x=5, 17
Hence, the points on the x-axis are (5, 0) and (17, 0).

Answer:

Let P (0, y) be a point on the y-axis. Then as per the question, we have
AP=BP0-62+y-52=0+42+y-3262+y-52=42+y-3262+y-52=42+y-32                     Squaring both sides
36+y2-10y+25=16+y2-6y+94y=36y=9
Hence, the point on the y-axis is (0, 9).

Page No 312:

Question 11:

Let P (0, y) be a point on the y-axis. Then as per the question, we have
AP=BP0-62+y-52=0+42+y-3262+y-52=42+y-3262+y-52=42+y-32                     Squaring both sides
36+y2-10y+25=16+y2-6y+94y=36y=9
Hence, the point on the y-axis is (0, 9).

Answer:

As per the question, we have
AP=BPx-52+y-12=x+12+y-52x-52+y-12=x+12+y-52        Squaring both sidesx2-10x+25+y2-2y+1=x2+2x+1+y2-10y+25
-10x-2y=2x-10y8y=12x3x=2y
Hence, 3x = 2y.

Page No 312:

Question 12:

As per the question, we have
AP=BPx-52+y-12=x+12+y-52x-52+y-12=x+12+y-52        Squaring both sidesx2-10x+25+y2-2y+1=x2+2x+1+y2-10y+25
-10x-2y=2x-10y8y=12x3x=2y
Hence, 3x = 2y.

Answer:

The given points are A(6, −1) and B(2, 3). The point P(x, y) is equidistant from the points A and B. So, PA = PB.
Also, (PA)2 = (PB)2
6-x2+-1-y2 = 2-x2+3-y2x2-12x+36+y2+2y+1 = x2-4x+4+y2-6y+9x2+y2-12x+2y+37 = x2+y2-4x-6y+13x2+y2-12x+2y-x2-y2+4x+6y =13-37-8x+8y = -24-8x-y = -24x-y =-24-8x-y =3

Hence proved.

Page No 312:

Question 13:

The given points are A(6, −1) and B(2, 3). The point P(x, y) is equidistant from the points A and B. So, PA = PB.
Also, (PA)2 = (PB)2
6-x2+-1-y2 = 2-x2+3-y2x2-12x+36+y2+2y+1 = x2-4x+4+y2-6y+9x2+y2-12x+2y+37 = x2+y2-4x-6y+13x2+y2-12x+2y-x2-y2+4x+6y =13-37-8x+8y = -24-8x-y = -24x-y =-24-8x-y =3

Hence proved.

Answer:

Let the required point be P(x, y). Then AP = BP = CP
That is, (AP)2=(BP)2=(CP)2
This means (AP)2=(BP)
x-52+y-32=x-52+y+52  x2-10x+25+y2-6y+9=x2-10x+25+y2+10y+25  x2-10x+y2-6y+34=x2-10x+y2+10y+50  x2-10x+y2-6y-x2+10x-y2-10y = 50-34  -16y = 16  y =-1616=-1  And BP2=CP2 x-52+y+52=x-12+y+52 x2-10x+25+y2+10y+25=x2-2x+1+y2+10y+25 x2-10x+y2+10y+50=x2-2x+y2+10y+26 x2-10x+y2+10y-x2+2x-y2-10y = 26-50 -8x = -24 x =-24-8= 3
Hence, the required point is (3, −1).

Page No 312:

Question 14:

Let the required point be P(x, y). Then AP = BP = CP
That is, (AP)2=(BP)2=(CP)2
This means (AP)2=(BP)
x-52+y-32=x-52+y+52  x2-10x+25+y2-6y+9=x2-10x+25+y2+10y+25  x2-10x+y2-6y+34=x2-10x+y2+10y+50  x2-10x+y2-6y-x2+10x-y2-10y = 50-34  -16y = 16  y =-1616=-1  And BP2=CP2 x-52+y+52=x-12+y+52 x2-10x+25+y2+10y+25=x2-2x+1+y2+10y+25 x2-10x+y2+10y+50=x2-2x+y2+10y+26 x2-10x+y2+10y-x2+2x-y2-10y = 26-50 -8x = -24 x =-24-8= 3
Hence, the required point is (3, −1).

Answer:

Given, the points A(4, 3) and B(x, 5) lie on a circle with centre O(2, 3).
Then OA = OB
Also (OA)2 = (OB)2
4-22+3-32 = x-22+5-3222+02 = x-22+224= x-22+4x-22 = 0x-2 = 0x=2
Therefore, x = 2.

Page No 312:

Question 15:

Given, the points A(4, 3) and B(x, 5) lie on a circle with centre O(2, 3).
Then OA = OB
Also (OA)2 = (OB)2
4-22+3-32 = x-22+5-3222+02 = x-22+224= x-22+4x-22 = 0x-2 = 0x=2
Therefore, x = 2.

Answer:

As per the question, we have
AC=BC-2-32+3+12=-2-x2+3-8252+42=x+22+-5225+16=x+22+25        Squaring both sides
25+16=x+22+25x+22=16x+2=±4x=-2±4=-2-4, -2+4=-6, 2
Now
BC=-2-x2+3-82      =-2-22+-5      =16+25=41 units
Hence, x = 2 or −6 and BC=41 units.

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Question 16:

As per the question, we have
AC=BC-2-32+3+12=-2-x2+3-8252+42=x+22+-5225+16=x+22+25        Squaring both sides
25+16=x+22+25x+22=16x+2=±4x=-2±4=-2-4, -2+4=-6, 2
Now
BC=-2-x2+3-82      =-2-22+-5      =16+25=41 units
Hence, x = 2 or −6 and BC=41 units.

Answer:

As per the question, we have
AP=BP2+22+2-k2=2+2k2+2+3242+2-k2=2+2k2+5216+4+k2-4k=4+4k2+8k+25        Squaring both sides
k2+4k+3=0k+1k+3=0k=-3, -1
Now for k=-1
AP=2+22+2-k2      =42+2+12                   =16+9=5 units
For k=-3
AP=2+22+2-k2      =42+2+32                   =16+25=41 units
Hence, k=-1,-3 ; AP=5 units for k=-1 and AP=41 units for k=-3.

Page No 312:

Question 17:

As per the question, we have
AP=BP2+22+2-k2=2+2k2+2+3242+2-k2=2+2k2+5216+4+k2-4k=4+4k2+8k+25        Squaring both sides
k2+4k+3=0k+1k+3=0k=-3, -1
Now for k=-1
AP=2+22+2-k2      =42+2+12                   =16+9=5 units
For k=-3
AP=2+22+2-k2      =42+2+32                   =16+25=41 units
Hence, k=-1,-3 ; AP=5 units for k=-1 and AP=41 units for k=-3.

Answer:

(i) As per the question, we have
x-a-b2+y-b+a2=x-a+b2+y-a-b2x-a-b2+y-b+a2=x-a+b2+y-a-b2      Squaring both sidesx2+a+b2-2xa+b+y2+a-b2-2ya-b=x2+a-b2-2xa-b+y2+a+b2-2ya+b-xa+b-ya-b=-xa-b-ya+b
-xa-xb-ay+by=-xa+bx-ya-byby=bx
Hence, bx = ay.

(ii)
As per the question, we have
AP=BPx-52+y-12=x+12+y-52x-52+y-12=x+12+y-52        Squaring both sidesx2-10x+25+y2-2y+1=x2+2x+1+y2-10y+25
-10x-2y=2x-10y8y=12x3x=2y
Hence, 3x = 2y.

Page No 312:

Question 18:

(i) As per the question, we have
x-a-b2+y-b+a2=x-a+b2+y-a-b2x-a-b2+y-b+a2=x-a+b2+y-a-b2      Squaring both sidesx2+a+b2-2xa+b+y2+a-b2-2ya-b=x2+a-b2-2xa-b+y2+a+b2-2ya+b-xa+b-ya-b=-xa-b-ya+b
-xa-xb-ay+by=-xa+bx-ya-byby=bx
Hence, bx = ay.

(ii)
As per the question, we have
AP=BPx-52+y-12=x+12+y-52x-52+y-12=x+12+y-52        Squaring both sidesx2-10x+25+y2-2y+1=x2+2x+1+y2-10y+25
-10x-2y=2x-10y8y=12x3x=2y
Hence, 3x = 2y.

Answer:

(i)
Let A(1, −1), B(5, 2) and C(9, 5) be the given points. Then
AB=5-12+2+12=42+32=25=5 unitsBC=9-52+5-22=42+32=25=5 unitsAC=9-12+5+12=82+62=100=10 units
AB+BC=5+5 units=10 units=AC
Hence, the given points are collinear.

(ii)
Let A(6, 9), B(0, 1) and C(−6, −7) be the given points. Then
AB=0-62+1-92=-62+-82=100=10 unitsBC=-6-02+-7-12=-62+-82=100=10 unitsAC=-6-62+-7-92=-122+-162=400=20 units
AB+BC=10+10 units=20 units=AC
Hence, the given points are collinear.

(iii)
Let A(−1, −1), B(2, 3) and C(8, 11) be the given points. Then
AB=2+12+3+12=32+42=25=5 unitsBC=8-22+11-32=62+82=100=10 unitsAC=8+12+11+12=92+122=225=15 units
AB+BC=5+10 units=15 units=AC
Hence, the given points are collinear.

(iv)
Let A(−2, 5), B(0, 1) and C(2, −3) be the given points. Then
AB=0+22+1-52=22+-42=20=25 unitsBC=2-02+-3-12=22+-42=20=25 unitsAC=2+22+-3-52=42+-82=80=45 units
AB+BC=25+25 units=45 units=AC
Hence, the given points are collinear.

Page No 312:

Question 19:

(i)
Let A(1, −1), B(5, 2) and C(9, 5) be the given points. Then
AB=5-12+2+12=42+32=25=5 unitsBC=9-52+5-22=42+32=25=5 unitsAC=9-12+5+12=82+62=100=10 units
AB+BC=5+5 units=10 units=AC
Hence, the given points are collinear.

(ii)
Let A(6, 9), B(0, 1) and C(−6, −7) be the given points. Then
AB=0-62+1-92=-62+-82=100=10 unitsBC=-6-02+-7-12=-62+-82=100=10 unitsAC=-6-62+-7-92=-122+-162=400=20 units
AB+BC=10+10 units=20 units=AC
Hence, the given points are collinear.

(iii)
Let A(−1, −1), B(2, 3) and C(8, 11) be the given points. Then
AB=2+12+3+12=32+42=25=5 unitsBC=8-22+11-32=62+82=100=10 unitsAC=8+12+11+12=92+122=225=15 units
AB+BC=5+10 units=15 units=AC
Hence, the given points are collinear.

(iv)
Let A(−2, 5), B(0, 1) and C(2, −3) be the given points. Then
AB=0+22+1-52=22+-42=20=25 unitsBC=2-02+-3-12=22+-42=20=25 unitsAC=2+22+-3-52=42+-82=80=45 units
AB+BC=25+25 units=45 units=AC
Hence, the given points are collinear.

Answer:

The given points are A(7, 10), B(−2, 5) and C(3, −4).
AB = -2-72+5-102 = -92+-52 =81+25 =106BC = 3--22+-4-52 = 52+-92 =25+81 = 106AC = 3-72+-4-102 = -42+-142 = 16+196 =212
Since, AB and BC are equal, they form the vertices of an isosceles triangle.
Also, (AB)2+(BC)21062+ 1062=212 
and (AC)2 = 2122 = 212
Thus, (AB)2+(BC)2 = (AC)2
This show that ABC is right- angled at B.
Therefore, the points A(7, 10), B(−2, 5) and C(3, −4) are the vertices of an isosceles right-angled triangle.

Page No 312:

Question 20:

The given points are A(7, 10), B(−2, 5) and C(3, −4).
AB = -2-72+5-102 = -92+-52 =81+25 =106BC = 3--22+-4-52 = 52+-92 =25+81 = 106AC = 3-72+-4-102 = -42+-142 = 16+196 =212
Since, AB and BC are equal, they form the vertices of an isosceles triangle.
Also, (AB)2+(BC)21062+ 1062=212 
and (AC)2 = 2122 = 212
Thus, (AB)2+(BC)2 = (AC)2
This show that ABC is right- angled at B.
Therefore, the points A(7, 10), B(−2, 5) and C(3, −4) are the vertices of an isosceles right-angled triangle.

Answer:

The given points are A(3, 0), B(6, 4) and C(− 1, 3). Now
AB=3-62+0-42=-32+-42     =9+16=25=5BC=6+12+4-32=72+12     =49+1=50=52
AC=3+12+0-32=42+-32     =16+9=25=5
AB=AC and AB2+AC2=BC2
Therefore, A(3, 0), B(6, 4) and C(− 1, 3) are the vertices of an isosceles right triangle.

Page No 312:

Question 21:

The given points are A(3, 0), B(6, 4) and C(− 1, 3). Now
AB=3-62+0-42=-32+-42     =9+16=25=5BC=6+12+4-32=72+12     =49+1=50=52
AC=3+12+0-32=42+-32     =16+9=25=5
AB=AC and AB2+AC2=BC2
Therefore, A(3, 0), B(6, 4) and C(− 1, 3) are the vertices of an isosceles right triangle.

Answer:

B=90AC2=AB2+BC25+22+2-t2=5-22+2+22+2+22+-2-t272+t-22=32+42+42+t+22
49+t2-4t+4=9+16+16+t2+4t+48-4t=4t8t=8t=1
Hence, t = 1.

Page No 312:

Question 22:

B=90AC2=AB2+BC25+22+2-t2=5-22+2+22+2+22+-2-t272+t-22=32+42+42+t+22
49+t2-4t+4=9+16+16+t2+4t+48-4t=4t8t=8t=1
Hence, t = 1.

Answer:

The given points are A(2, 4), B(2, 6) and C2+3, 5. Now
AB=2-22+4-62=02+-22     =0+4=2BC=2-2-32+6-52=-32+12     =3+1=2
AC=2-2-32+4-52=-32+-12     =3+1=2
Hence, the points A(2, 4), B(2, 6) and C2+3, 5 are the vertices of an equilateral triangle.

Page No 312:

Question 23:

The given points are A(2, 4), B(2, 6) and C2+3, 5. Now
AB=2-22+4-62=02+-22     =0+4=2BC=2-2-32+6-52=-32+12     =3+1=2
AC=2-2-32+4-52=-32+-12     =3+1=2
Hence, the points A(2, 4), B(2, 6) and C2+3, 5 are the vertices of an equilateral triangle.

Answer:

Let the given points be A(− 3, − 3), B(3, 3) and C-33, 33. Now
AB=-3-32+-3-32=-62+-62     =36+36=72=62BC=3+332+3-332      =9+27+183+9+27-183=72=62
AC=-3+332+-3-332=3-332+3+332     =9+27-183+9+27+183      =72=62
Hence, the given points are the vertices of an equilateral triangle.

Page No 312:

Question 24:

Let the given points be A(− 3, − 3), B(3, 3) and C-33, 33. Now
AB=-3-32+-3-32=-62+-62     =36+36=72=62BC=3+332+3-332      =9+27+183+9+27-183=72=62
AC=-3+332+-3-332=3-332+3+332     =9+27-183+9+27+183      =72=62
Hence, the given points are the vertices of an equilateral triangle.

Answer:

Let the given points be A(−5, 6) B(3, 0) and C(9, 8).
AB = 3--52+0-62 = 82+-62 =64+36 =100 =10 unitsBC = 9-32+8-02 = 62+82 =36+64 = 100 = 10 unitsAC = 9--52+8-62 = 142+22 = 196+4 =200 = 102 unitsTherefore, AB= BC = 10 units

Also, (AB)2+(BC)2102+ 102=200 
and (AC)2 = 1022=200
Thus, (AB)2+(BC)2 = (AC)2
This show that ABC is right- angled at B.
Therefore, the points A(−5, 6) B(3, 0) and C(9, 8) are the vertices of an isosceles right-angled triangle.
Also, area of a triangle = 12×base×height
If AB is the height and BC is the base,Area = 12×10×10 = 50 square units



Page No 313:

Question 25:

Let the given points be A(−5, 6) B(3, 0) and C(9, 8).
AB = 3--52+0-62 = 82+-62 =64+36 =100 =10 unitsBC = 9-32+8-02 = 62+82 =36+64 = 100 = 10 unitsAC = 9--52+8-62 = 142+22 = 196+4 =200 = 102 unitsTherefore, AB= BC = 10 units

Also, (AB)2+(BC)2102+ 102=200 
and (AC)2 = 1022=200
Thus, (AB)2+(BC)2 = (AC)2
This show that ABC is right- angled at B.
Therefore, the points A(−5, 6) B(3, 0) and C(9, 8) are the vertices of an isosceles right-angled triangle.
Also, area of a triangle = 12×base×height
If AB is the height and BC is the base,Area = 12×10×10 = 50 square units

Answer:

The given points are O(0, 0) A(3, 3) and B(3, − 3).
OA = 3-02+3-02 = 32+32 = 9+3 = 12 =23 unitsAB = 3-32+-3 -32 = 0+232 =43 = 12 = 23 unitsOB = 3-02+-3 -02 =32+32 = 9+3 =12 =23 unitsTherefore, OA = AB = OB = 23 units
Thus, t
he points O(0, 0) A(3, 3)and B(3, − 3) are the vertices of an equilateral triangle.
Also, the area of the triangle OAB = 34×side2
34×232  =34×12  = 33 square units

Page No 313:

Question 26:

The given points are O(0, 0) A(3, 3) and B(3, − 3).
OA = 3-02+3-02 = 32+32 = 9+3 = 12 =23 unitsAB = 3-32+-3 -32 = 0+232 =43 = 12 = 23 unitsOB = 3-02+-3 -02 =32+32 = 9+3 =12 =23 unitsTherefore, OA = AB = OB = 23 units
Thus, t
he points O(0, 0) A(3, 3)and B(3, − 3) are the vertices of an equilateral triangle.
Also, the area of the triangle OAB = 34×side2
34×232  =34×12  = 33 square units

Answer:

(i) The given points are A(3, 2), B(0, 5), C(−3, 2) and D(0, −1).
AB = 0-32+5-22 = -32+32 = 9+9 = 18 =32 unitsBC = -3-02+2-52 = -32+-32 = 9+9 = 18 =32 unitsCD = 0+32+-1-22 = 32+-32 = 9+9 = 18 =32 unitsDA =0-32+-1-22 = -32+-32 = 9+9 = 18 =32 unitsTherefore, AB = BC = CD = DA = 32 unitsAlso, AC = -3-32+2-22 = -62+02 = 36 = 6 unitsBD = 0-02+-1-52 = 02+-62 = 36 = 6 unitsThus, diagonal AC = diagonal BD
Therefore, the given points form a square.


(ii) The given points are A(6, 2), B(2, 1), C(1, 5) and D(5, 6).
AB = 2-62+1-22 = -42+-12 = 16+1 = 17 unitsBC = 1-22+5-12 = -12+-42 = 1+16 = 17 unitsCD = 5-12+6-52 = 42+12 = 16+1 = 17 unitsDA =5-62+6-22 = 12+42 = 1+16 = 17 unitsTherefore AB = BC = CD = DA = 17 unitsAlso AC = 1-62+5-22 = -52+32 = 25+9 = 34 unitsBD = 5-22+6-12 = 32+52 = 9+25 = 34 unitsThus, diagonal AC = diagonal BD
Therefore, the given points form a square.


(iii) The given points are P(0, −2), Q(3, 1), R(0, 4) and S(−3, 1).
PQ = 3-02+1+22 = 32+32 = 9+9 = 18 = 32 unitsQR = 0-32+4-12 = -32+32 = 9+9 = 18 =32 unitsRS = -3-02+1-42 = -32+-32 = 9+9 = 18 = 32unitsSP =-3-02+1+22 = -32+32 = 9+9 = 18  = 32unitsTherefore, PQ = QR = RS = SP = 32 unitsAlso, PR = 0-02+4+22 = 02+62 = 36 = 6 unitsQS= -3-32+1-12 = -62+02 = 36 = 6 unitsThus, diagonal PR = diagonal QS
Therefore, the given points form a square.

Page No 313:

Question 27:

(i) The given points are A(3, 2), B(0, 5), C(−3, 2) and D(0, −1).
AB = 0-32+5-22 = -32+32 = 9+9 = 18 =32 unitsBC = -3-02+2-52 = -32+-32 = 9+9 = 18 =32 unitsCD = 0+32+-1-22 = 32+-32 = 9+9 = 18 =32 unitsDA =0-32+-1-22 = -32+-32 = 9+9 = 18 =32 unitsTherefore, AB = BC = CD = DA = 32 unitsAlso, AC = -3-32+2-22 = -62+02 = 36 = 6 unitsBD = 0-02+-1-52 = 02+-62 = 36 = 6 unitsThus, diagonal AC = diagonal BD
Therefore, the given points form a square.


(ii) The given points are A(6, 2), B(2, 1), C(1, 5) and D(5, 6).
AB = 2-62+1-22 = -42+-12 = 16+1 = 17 unitsBC = 1-22+5-12 = -12+-42 = 1+16 = 17 unitsCD = 5-12+6-52 = 42+12 = 16+1 = 17 unitsDA =5-62+6-22 = 12+42 = 1+16 = 17 unitsTherefore AB = BC = CD = DA = 17 unitsAlso AC = 1-62+5-22 = -52+32 = 25+9 = 34 unitsBD = 5-22+6-12 = 32+52 = 9+25 = 34 unitsThus, diagonal AC = diagonal BD
Therefore, the given points form a square.


(iii) The given points are P(0, −2), Q(3, 1), R(0, 4) and S(−3, 1).
PQ = 3-02+1+22 = 32+32 = 9+9 = 18 = 32 unitsQR = 0-32+4-12 = -32+32 = 9+9 = 18 =32 unitsRS = -3-02+1-42 = -32+-32 = 9+9 = 18 = 32unitsSP =-3-02+1+22 = -32+32 = 9+9 = 18  = 32unitsTherefore, PQ = QR = RS = SP = 32 unitsAlso, PR = 0-02+4+22 = 02+62 = 36 = 6 unitsQS= -3-32+1-12 = -62+02 = 36 = 6 unitsThus, diagonal PR = diagonal QS
Therefore, the given points form a square.

Answer:

The given points are A(−3, 2), B(−5, −5), C(2, −3) and D(4, 4).
AB = -5+32+-5-22 = -22+-72 = 4+49 = 53 units.BC = 2+52+-3+52 = 72+22 = 49+4 = 53 units.CD = 4-22+4+32 = 22+72 = 4+49 = 53 units.DA= 4+32+4-22 = 72+22 = 49+4 = 53 units.Therefore, AB= BC= CD = DA = 53 unitsAC = 2+32+-3-22 = 52+-52 =25+25 = 50 =25×2= 52 unitsBD = 4+52+4+52 = 92+92 = 81+81 = 162 =81×2= 92 unitsThus, diagonal AC is not equal to diagonal BD.
Therefore, ABCD is a quadrilateral with equal sides and unequal diagonals.
Hence, ABCD is a rhombus.
Area of a rhombus = 12×product of diagonals                                    = 12×52×92                                    = 4522                                    = 45 square units

Page No 313:

Question 28:

The given points are A(−3, 2), B(−5, −5), C(2, −3) and D(4, 4).
AB = -5+32+-5-22 = -22+-72 = 4+49 = 53 units.BC = 2+52+-3+52 = 72+22 = 49+4 = 53 units.CD = 4-22+4+32 = 22+72 = 4+49 = 53 units.DA= 4+32+4-22 = 72+22 = 49+4 = 53 units.Therefore, AB= BC= CD = DA = 53 unitsAC = 2+32+-3-22 = 52+-52 =25+25 = 50 =25×2= 52 unitsBD = 4+52+4+52 = 92+92 = 81+81 = 162 =81×2= 92 unitsThus, diagonal AC is not equal to diagonal BD.
Therefore, ABCD is a quadrilateral with equal sides and unequal diagonals.
Hence, ABCD is a rhombus.
Area of a rhombus = 12×product of diagonals                                    = 12×52×92                                    = 4522                                    = 45 square units

Answer:

The given points are A(3, 0), B(4, 5), C(− 1, 4) and D(− 2, − 1).
AB=3-42+0-52=-12+-52     =1+25=26
BC=4+12+5-42=52+12     =25+1=26
CD=-1+22+4+12=12+52     =1+25=26AD=3+22+0+12=52+12     =25+1=26
AC=3+12+0-42=42+-42     =16+16=42BD=4+22+5+12=62+62     =36+36=62
AB=BC=CD=AD=62 and ACBD
Therefore, the given points are the vertices of a rhombus.
AreaABCD=12×AC×BD                          =12×42×62=24 sq. units
Hence, the area of the rhombus is 24 sq. units.

Page No 313:

Question 29:

The given points are A(3, 0), B(4, 5), C(− 1, 4) and D(− 2, − 1).
AB=3-42+0-52=-12+-52     =1+25=26
BC=4+12+5-42=52+12     =25+1=26
CD=-1+22+4+12=12+52     =1+25=26AD=3+22+0+12=52+12     =25+1=26
AC=3+12+0-42=42+-42     =16+16=42BD=4+22+5+12=62+62     =36+36=62
AB=BC=CD=AD=62 and ACBD
Therefore, the given points are the vertices of a rhombus.
AreaABCD=12×AC×BD                          =12×42×62=24 sq. units
Hence, the area of the rhombus is 24 sq. units.

Answer:

The given points are A(6, 1), B(8, 2), C(9, 4) and D(7, 3).
AB=6-82+1-22=-22+-12     =4+1=5
BC=8-92+2-42=-12+-22     =1+4=5
CD=9-72+4-32=22+12     =4+1=5AD=7-62+3-12=12+22     =1+4=5
AC=6-92+1-42=-32+-32     =9+9=32BD=8-72+2-32=12+-12     =1+1=2
AB=BC=CD=AD=5 and ACBD
Therefore, the given points are the vertices of a rhombus. Now
AreaABCD=12×AC×BD                          =12×32×2=3 sq. units
Hence, the area of the rhombus is 3 sq. units.

Page No 313:

Question 30:

The given points are A(6, 1), B(8, 2), C(9, 4) and D(7, 3).
AB=6-82+1-22=-22+-12     =4+1=5
BC=8-92+2-42=-12+-22     =1+4=5
CD=9-72+4-32=22+12     =4+1=5AD=7-62+3-12=12+22     =1+4=5
AC=6-92+1-42=-32+-32     =9+9=32BD=8-72+2-32=12+-12     =1+1=2
AB=BC=CD=AD=5 and ACBD
Therefore, the given points are the vertices of a rhombus. Now
AreaABCD=12×AC×BD                          =12×32×2=3 sq. units
Hence, the area of the rhombus is 3 sq. units.

Answer:

The given points are A(2, 1), B(5, 2), C(6, 4) and D(3, 3).
AB = 5-22+2-12 = 32+12 = 9+1 =10 unitsBC = 6-52+4-22 = 12+22 = 1+4 =5 unitsCD = 3-62+3-42 =-32+-12 = 9+1 =10 unitsAD = 3-22+3-12 = 12+22 = 1+4 =5 unitsThus, AB = CD = 10 units and BC = AD =5 unitsSo, quadrilateral ABCD is a parallelogramAlso, AC = 6-22+4-12 = 42+32 = 16+9 = 25=5 unitsBD = 3-52+3-22 =-22+12 = 4+1 = 5 units
But diagonal AC is not equal to diagonal BD.
Hence, the given points do not form a rectangle.

Page No 313:

Question 31:

The given points are A(2, 1), B(5, 2), C(6, 4) and D(3, 3).
AB = 5-22+2-12 = 32+12 = 9+1 =10 unitsBC = 6-52+4-22 = 12+22 = 1+4 =5 unitsCD = 3-62+3-42 =-32+-12 = 9+1 =10 unitsAD = 3-22+3-12 = 12+22 = 1+4 =5 unitsThus, AB = CD = 10 units and BC = AD =5 unitsSo, quadrilateral ABCD is a parallelogramAlso, AC = 6-22+4-12 = 42+32 = 16+9 = 25=5 unitsBD = 3-52+3-22 =-22+12 = 4+1 = 5 units
But diagonal AC is not equal to diagonal BD.
Hence, the given points do not form a rectangle.

Answer:

The given vertices are A(1, 2), B(4, 3), C(6, 6) and D(3, 5).
AB=1-42+2-32=-32+-12     =9+1=10
BC=4-62+3-62=-22+-32     =4+9=13
CD=6-32+6-52=32+12     =9+1=10AD=1-32+2-52=-22+-32     =4+9=13
AB=CD=10 units and BC=AD=13 units
Therefore, ABCD is a parallelogram. Now
AC=1-62+2-62=-52+-42     =25+16=41BD=4-32+3-52=12+-22     =1+4=5
Thus, the diagonals AC and BD are not equal and hence ABCD is not a rectangle.

Page No 313:

Question 32:

The given vertices are A(1, 2), B(4, 3), C(6, 6) and D(3, 5).
AB=1-42+2-32=-32+-12     =9+1=10
BC=4-62+3-62=-22+-32     =4+9=13
CD=6-32+6-52=32+12     =9+1=10AD=1-32+2-52=-22+-32     =4+9=13
AB=CD=10 units and BC=AD=13 units
Therefore, ABCD is a parallelogram. Now
AC=1-62+2-62=-52+-42     =25+16=41BD=4-32+3-52=12+-22     =1+4=5
Thus, the diagonals AC and BD are not equal and hence ABCD is not a rectangle.

Answer:

(i)
Given: A-4,-1,B-2,-4,C4,0 and D2,3In rectangle the opposite sides are equalUsing d=x2-x12+y2-y12AB=-4+22+-1+42AB=13BC=4+22+0+42BC=52CD=2-42+3-02CD=13AD=2+42+3+12AD=52AB=CD and BC=AD
Now, we will prove for rectangles because diagnols in a rectangle are also equalAC=4--42+0--12AC=4+42+12AC=65AndBD=2--22+3--42BD=42+72BD=65We can see that diagonals are also equal Therefore, given coordinates are of rectangle.

(ii) The given points are A(2, −2), B(14, 10) C(11, 13) and D(−1, 1).
AB = 14-22+10--22 = 122+122 = 144+144 =288 =122 unitsBC= 11-142+13-102 = -32+32 = 9+9 = 18 =32 unitsCD = -1-112+1-132 = -122+-122 = 144+144 =288 =122 unitsAD = -1-22+1--22 = -32+32 = 9+9 =18 =32 unitsThus, AB = CD = 122 units and BC = AD =32 unitsAlso, AC = 11-22+13--22 = 92+152 = 81+225 =306 =334 unitsBD = -1-142+1-102 = -152+-92 = 81+225 = 306=334 units
Also, diagonal AC = diagonal BD
Hence, the given points form a rectangle.

(iii) The given points are A(0, −4), B(6, 2) C(3, 5) and D(−3, −1).
AB = 6-02+2--42 = 62+62 = 36+36 =72 =62 unitsBC = 3-62+5-22 = -32+32 = 9+9 =18=32 unitsCD = -3-32+-1-52 = -62+-62 = 36+36 =72 =62 unitsAD = -3-02+-1--42 = -32+32 = 9+9 =18 =32 unitsThus,  AB = CD = 10 units and BC = AD =5 unitsAlso, AC = 3-02+5--42 = 32+92 = 9+81 =90=310 unitsBD = -3-62+-1-22 = -92+-32 = 81+9 = 90=310 units
Also, diagonal AC = diagonal BD.
Hence, the given points form a rectangle.

Page No 313:

Question 33:

(i)
Given: A-4,-1,B-2,-4,C4,0 and D2,3In rectangle the opposite sides are equalUsing d=x2-x12+y2-y12AB=-4+22+-1+42AB=13BC=4+22+0+42BC=52CD=2-42+3-02CD=13AD=2+42+3+12AD=52AB=CD and BC=AD
Now, we will prove for rectangles because diagnols in a rectangle are also equalAC=4--42+0--12AC=4+42+12AC=65AndBD=2--22+3--42BD=42+72BD=65We can see that diagonals are also equal Therefore, given coordinates are of rectangle.

(ii) The given points are A(2, −2), B(14, 10) C(11, 13) and D(−1, 1).
AB = 14-22+10--22 = 122+122 = 144+144 =288 =122 unitsBC= 11-142+13-102 = -32+32 = 9+9 = 18 =32 unitsCD = -1-112+1-132 = -122+-122 = 144+144 =288 =122 unitsAD = -1-22+1--22 = -32+32 = 9+9 =18 =32 unitsThus, AB = CD = 122 units and BC = AD =32 unitsAlso, AC = 11-22+13--22 = 92+152 = 81+225 =306 =334 unitsBD = -1-142+1-102 = -152+-92 = 81+225 = 306=334 units
Also, diagonal AC = diagonal BD
Hence, the given points form a rectangle.

(iii) The given points are A(0, −4), B(6, 2) C(3, 5) and D(−3, −1).
AB = 6-02+2--42 = 62+62 = 36+36 =72 =62 unitsBC = 3-62+5-22 = -32+32 = 9+9 =18=32 unitsCD = -3-32+-1-52 = -62+-62 = 36+36 =72 =62 unitsAD = -3-02+-1--42 = -32+32 = 9+9 =18 =32 unitsThus,  AB = CD = 10 units and BC = AD =5 unitsAlso, AC = 3-02+5--42 = 32+92 = 9+81 =90=310 unitsBD = -3-62+-1-22 = -92+-32 = 81+9 = 90=310 units
Also, diagonal AC = diagonal BD.
Hence, the given points form a rectangle.

Answer:

In ΔABC, the coordinates of the vertices are A(–2, 0), B(0,2), C(2,0). 

AB=0+22+2-02=8=22CB=0-22+2-02=8=22AC=2+22+0-02=4

In ΔDEF, the coordinates of the vertices are D(–4, 0), E(4, 0), F(0, 4).

DF=4+42+0-02=8FE=0-42+4-02=42DE=0+42+4-02=42

Now, for ΔABC and ΔDEF to be similar, the corresponding sides should be proportional. 

So, ACDF=BCFE=ABDE48=2242=224212=12=12Since, the corresponding sides are proportional Therefore, given two triangles are similar.



Page No 324:

Question 1:

In ΔABC, the coordinates of the vertices are A(–2, 0), B(0,2), C(2,0). 

AB=0+22+2-02=8=22CB=0-22+2-02=8=22AC=2+22+0-02=4

In ΔDEF, the coordinates of the vertices are D(–4, 0), E(4, 0), F(0, 4).

DF=4+42+0-02=8FE=0-42+4-02=42DE=0+42+4-02=42

Now, for ΔABC and ΔDEF to be similar, the corresponding sides should be proportional. 

So, ACDF=BCFE=ABDE48=2242=224212=12=12Since, the corresponding sides are proportional Therefore, given two triangles are similar.

Answer:

(i) The end points of AB are A(−1, 7) and B(4, −3).
Therefore, (x1 = −1, y1 = 7) and (x2 = 4, y2 = −3)
Also, m = 2 and n = 3
Let the required point be P(x, y).
By section formula, we get:
x =mx2+nx1m+n, y = my2+ny1m+nx= 2×4 + 3×-12+3, y =2×-3+3×72+3x= 8-35, y = -6+215x = 55, y = 155Therefore, x = 1 and y = 3
Hence, the coordinates of the required point are (1, 3).

(ii) The end points of AB are A(−5, 11) and B(4, −7).
Therefore, (x1 = −5, y1 = 11) and (x2 = 4, y2 = −7).
Also, m = 7 and n = 2
Let the required point be P(x, y).
By section formula, we have:
x =mx2+nx1m+n, y = my2+ny1m+nx= 7×4 + 2×-57+2, y =7×-7+2×117+2x= 28-109, y = -49+229x = 189, y = -279Therefore, x = 2 and y = -3
Hence, the required point is P(2, −3).

Page No 324:

Question 2:

(i) The end points of AB are A(−1, 7) and B(4, −3).
Therefore, (x1 = −1, y1 = 7) and (x2 = 4, y2 = −3)
Also, m = 2 and n = 3
Let the required point be P(x, y).
By section formula, we get:
x =mx2+nx1m+n, y = my2+ny1m+nx= 2×4 + 3×-12+3, y =2×-3+3×72+3x= 8-35, y = -6+215x = 55, y = 155Therefore, x = 1 and y = 3
Hence, the coordinates of the required point are (1, 3).

(ii) The end points of AB are A(−5, 11) and B(4, −7).
Therefore, (x1 = −5, y1 = 11) and (x2 = 4, y2 = −7).
Also, m = 7 and n = 2
Let the required point be P(x, y).
By section formula, we have:
x =mx2+nx1m+n, y = my2+ny1m+nx= 7×4 + 2×-57+2, y =7×-7+2×117+2x= 28-109, y = -49+229x = 189, y = -279Therefore, x = 2 and y = -3
Hence, the required point is P(2, −3).

Answer:


Consider the figure.
Here points P and Q trisect AB.
Therefore, P divides AB into 1 : 2 and Q divides AB into 2 : 1.
Using section formula, coordinates of P are;
P(x,y)=1×1+2×73,1×-5+2×-23P(x,y)=153,-93=5,-3
Similarly, coordinates of Q are;
Q(a,b)=2×1+1×73,2×-5+1×-23Q(a,b)=93,-123=3,-4
Therefore, coordinates of points P and Q are (5, -3) and (3, -4) respectively.

Page No 324:

Question 3:


Consider the figure.
Here points P and Q trisect AB.
Therefore, P divides AB into 1 : 2 and Q divides AB into 2 : 1.
Using section formula, coordinates of P are;
P(x,y)=1×1+2×73,1×-5+2×-23P(x,y)=153,-93=5,-3
Similarly, coordinates of Q are;
Q(a,b)=2×1+1×73,2×-5+1×-23Q(a,b)=93,-123=3,-4
Therefore, coordinates of points P and Q are (5, -3) and (3, -4) respectively.

Answer:

The coordinates of the points A and B are (−2, −2) and (2, −4) respectively, where AP=37AB and P lies on the line segment AB. So
AP+BP=ABAP+BP=7AP3                 AP=37ABBP=7AP3-APAPBP=34
Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally. Then
x=3×2+4×-23+4=6-87=-27y=3×-4+4×-23+4=-12-87=-207
Hence, the coordinates of point P are -27, -207.



Page No 325:

Question 4:

The coordinates of the points A and B are (−2, −2) and (2, −4) respectively, where AP=37AB and P lies on the line segment AB. So
AP+BP=ABAP+BP=7AP3                 AP=37ABBP=7AP3-APAPBP=34
Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally. Then
x=3×2+4×-23+4=6-87=-27y=3×-4+4×-23+4=-12-87=-207
Hence, the coordinates of point P are -27, -207.

Answer:

Let the coordinates of A be (x, y). Here, PAPQ=25. So,
PA+AQ=PQPA+AQ=5PA2                 [PA=25PQ]AQ=5PA2-PAAQPA=32PAAQ=23
Let (x, y) be the coordinates of A, which divides PQ in the ratio 2 : 3 internally. Then using section formula, we get
x=2×-4+3×62+3=-8+185=105=2y=2×-1+3×-62+3=-2-185=-205=-4
Now, the point (2, −4) lies on the line 3x + k (y + 1) = 0, therefore
3×2+k-4+1=03k=6k=63=2
Hence, k = 2.

Page No 325:

Question 5:

Let the coordinates of A be (x, y). Here, PAPQ=25. So,
PA+AQ=PQPA+AQ=5PA2                 [PA=25PQ]AQ=5PA2-PAAQPA=32PAAQ=23
Let (x, y) be the coordinates of A, which divides PQ in the ratio 2 : 3 internally. Then using section formula, we get
x=2×-4+3×62+3=-8+185=105=2y=2×-1+3×-62+3=-2-185=-205=-4
Now, the point (2, −4) lies on the line 3x + k (y + 1) = 0, therefore
3×2+k-4+1=03k=6k=63=2
Hence, k = 2.

Answer:

Since, the points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in five equal parts, so
AP = PQ = QR = RS = SB
Here, point P divides AB in the ratio of 1 : 4 internally. So using section formula, we get
Coordinates of P=1×6+4×11+4, 1×7+4×21+4                           =6+45, 7+85=2, 3
The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get
Coordinates of Q=2×6+3×12+3, 2×7+3×22+3                           =12+35, 14+65=3, 4
The point R divides AB in the ratio of 3 : 2 internally. So using section formula, we get
Coordinates of R=3×6+2×13+2, 3×7+2×23+2                           =18+25, 21+45=4, 5
Hence, the coordinates of the points P, Q and R are (2, 3), (3, 4) and (4, 5) respectively.

Page No 325:

Question 6:

Since, the points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in five equal parts, so
AP = PQ = QR = RS = SB
Here, point P divides AB in the ratio of 1 : 4 internally. So using section formula, we get
Coordinates of P=1×6+4×11+4, 1×7+4×21+4                           =6+45, 7+85=2, 3
The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get
Coordinates of Q=2×6+3×12+3, 2×7+3×22+3                           =12+35, 14+65=3, 4
The point R divides AB in the ratio of 3 : 2 internally. So using section formula, we get
Coordinates of R=3×6+2×13+2, 3×7+2×23+2                           =18+25, 21+45=4, 5
Hence, the coordinates of the points P, Q and R are (2, 3), (3, 4) and (4, 5) respectively.

Answer:

The given points are A(1, 6) and B(5, −2).
Then, P(x, y) is a point that divides the line AB in the ratio 1:3.
By the section formula:
x =mx2+nx1m+n, y = my2+ny1m+nx= 1×5 + 3×11+3, y =1×-2+3×61+3x= 5+34, y = -2+184x = 84, y = 164 x = 2 and y = 4
Therefore, the coordinates of point P are (2, 4).
Let Q be the mid point of AB.
Then, Q(x, y):
x = x1+x22, y =y1+y22x = 1+52, y =6+-22x = 62, y = 42x = 3, y = 2
Therefore, the coordinates of Q are (3, 2).
Let R (x, y) be a point that divides AB in the ratio 3:1.
Then, by the section formula:
x =mx2+nx1m+n, y = my2+ny1m+nx= 3×5 + 1×13+1, y =3×-2+1×63+1x= 15+14, y = -6+64x = 164, y = 04 x = 4 and y = 0
Therefore, the coordinates of R are (4, 0).
Hence, the coordinates of point P, Q and R are (2, 4), (3, 2) and (4, 0) respectively.

Page No 325:

Question 7:

The given points are A(1, 6) and B(5, −2).
Then, P(x, y) is a point that divides the line AB in the ratio 1:3.
By the section formula:
x =mx2+nx1m+n, y = my2+ny1m+nx= 1×5 + 3×11+3, y =1×-2+3×61+3x= 5+34, y = -2+184x = 84, y = 164 x = 2 and y = 4
Therefore, the coordinates of point P are (2, 4).
Let Q be the mid point of AB.
Then, Q(x, y):
x = x1+x22, y =y1+y22x = 1+52, y =6+-22x = 62, y = 42x = 3, y = 2
Therefore, the coordinates of Q are (3, 2).
Let R (x, y) be a point that divides AB in the ratio 3:1.
Then, by the section formula:
x =mx2+nx1m+n, y = my2+ny1m+nx= 3×5 + 1×13+1, y =3×-2+1×63+1x= 15+14, y = -6+64x = 164, y = 04 x = 4 and y = 0
Therefore, the coordinates of R are (4, 0).
Hence, the coordinates of point P, Q and R are (2, 4), (3, 2) and (4, 0) respectively.

Answer:

Let P and Q be the points of trisection of AB.
Then, P divides AB in the ratio 1:2.
So, the coordinates of P are
x =mx2+nx1m+n, y = my2+ny1m+nx= 1×1 + 2×31+2, y =1×2+2×-41+2x= 1+63, y = 2-83x= 73, y = -63x = 73, y = -2
Hence, the coordinates of P are (73, −2).
But (p, −2) are the coordinates of P.
So, p=73
Also, Q divides the line AB in the ratio 2:1.
So, the coordinates of Q are
x =mx2+nx1m+n, y = my2+ny1m+nx= 2×1 + 1×32+1, y =2×2+1×-42+1x= 2+33, y = 4-43x = 53, y = 0Hence, coordinates of Q are 53, 0.
But the given coordinates of Q are 53, q.
So, q = 0
Thus, p=73 and q=053

Page No 325:

Question 8:

Let P and Q be the points of trisection of AB.
Then, P divides AB in the ratio 1:2.
So, the coordinates of P are
x =mx2+nx1m+n, y = my2+ny1m+nx= 1×1 + 2×31+2, y =1×2+2×-41+2x= 1+63, y = 2-83x= 73, y = -63x = 73, y = -2
Hence, the coordinates of P are (73, −2).
But (p, −2) are the coordinates of P.
So, p=73
Also, Q divides the line AB in the ratio 2:1.
So, the coordinates of Q are
x =mx2+nx1m+n, y = my2+ny1m+nx= 2×1 + 1×32+1, y =2×2+1×-42+1x= 2+33, y = 4-43x = 53, y = 0Hence, coordinates of Q are 53, 0.
But the given coordinates of Q are 53, q.
So, q = 0
Thus, p=73 and q=053

Answer:

(i) The given points are A(3, 0) and B(−5, 4).
Let (x, y) be the mid point of AB. Then:
x = x1+x22, y =y1+y22x = 3+-52, y =0+42x = -22, y = 42x = -1, y = 2
Therefore, (−1, 2) are the coordinates of mid point of AB.

(ii) The given points are P(−11, −8) and Q(8, −2).
Let (x, y) be the mid point of PQ. Then:
x = x1+x22, y =y1+y22x = -11+82, y =-8-22x =- 32, y = -102x = -32, y = -5
Therefore, -32, -5 are the coordinates of midpoint of PQ.

Page No 325:

Question 9:

(i) The given points are A(3, 0) and B(−5, 4).
Let (x, y) be the mid point of AB. Then:
x = x1+x22, y =y1+y22x = 3+-52, y =0+42x = -22, y = 42x = -1, y = 2
Therefore, (−1, 2) are the coordinates of mid point of AB.

(ii) The given points are P(−11, −8) and Q(8, −2).
Let (x, y) be the mid point of PQ. Then:
x = x1+x22, y =y1+y22x = -11+82, y =-8-22x =- 32, y = -102x = -32, y = -5
Therefore, -32, -5 are the coordinates of midpoint of PQ.

Answer:

The given points are A(6, −5) and B(−2, 11).
Let (x, y) be the mid point of AB. Then:
x = x1+x22, y =y1+y22x = 6+-22, y =-5+112x = 6-22, y = -5+112x = 42, y = 62x = 2, y = 3
So, the midpoint of AB is (2, 3).
But it is given that the midpoint of AB is (2, p).
Therefore, the value of p = 3.

Page No 325:

Question 10:

The given points are A(6, −5) and B(−2, 11).
Let (x, y) be the mid point of AB. Then:
x = x1+x22, y =y1+y22x = 6+-22, y =-5+112x = 6-22, y = -5+112x = 42, y = 62x = 2, y = 3
So, the midpoint of AB is (2, 3).
But it is given that the midpoint of AB is (2, p).
Therefore, the value of p = 3.

Answer:

The points are A(2a, 4) and B(−2, 3b).
Let C(1, 2a + 1) be the mid point of AB. Then:
x = x1+x22, y =y1+y221 = 2a+-22, 2a+1 =4+3b22=2a-2,  4a+2 = 4+3b2a = 2+2  , 4a-3b = 4-2a=42, 4a-3b = 2a = 2, 4a-3b = 2Putting the value of a in the equation  4a+3b = 2, we get:42-3b = 2-3b = 2-8 = -6b = 63 = 2Therefore, a=2 and b=2.

Page No 325:

Question 11:

The points are A(2a, 4) and B(−2, 3b).
Let C(1, 2a + 1) be the mid point of AB. Then:
x = x1+x22, y =y1+y221 = 2a+-22, 2a+1 =4+3b22=2a-2,  4a+2 = 4+3b2a = 2+2  , 4a-3b = 4-2a=42, 4a-3b = 2a = 2, 4a-3b = 2Putting the value of a in the equation  4a+3b = 2, we get:42-3b = 2-3b = 2-8 = -6b = 63 = 2Therefore, a=2 and b=2.

Answer:

The given points are A(−2, 9) and B(6, 3).
Then, C(x, y) is the midpoint of AB.
x = x1+x22, y =y1+y22x = -2+62, y =9+32x = 42, y = 122x = 2, y = 6
Therefore, the coordinates of point C are (2, 6).

Page No 325:

Question 12:

The given points are A(−2, 9) and B(6, 3).
Then, C(x, y) is the midpoint of AB.
x = x1+x22, y =y1+y22x = -2+62, y =9+32x = 42, y = 122x = 2, y = 6
Therefore, the coordinates of point C are (2, 6).

Answer:

C(2, −3) is the centre of the given circle. Let A(a, b) and B(1, 4) be the two end-points of the given diameter AB. Then, the coordinates of C are
x = a+12, y =b+42It is given that x = 2 and y = -3.2 =a+12, -3 = b+424 = a+1 , -6 = b+4a = 4-1, b=-6-4a = 3, b= -10
Therefore, the coordinates of point A are (3, -10).

Page No 325:

Question 13:

C(2, −3) is the centre of the given circle. Let A(a, b) and B(1, 4) be the two end-points of the given diameter AB. Then, the coordinates of C are
x = a+12, y =b+42It is given that x = 2 and y = -3.2 =a+12, -3 = b+424 = a+1 , -6 = b+4a = 4-1, b=-6-4a = 3, b= -10
Therefore, the coordinates of point A are (3, -10).

Answer:

Let the point P(2, 5) divide AB in the ratio k : 1.
Then, by section formula, the coordinates of P are
x=-6k+8k+1, y=9k+2k+1It is given that the coordinates of P are P(2, 5).2 = -6k+8k+1, 5 =9k+2k+12k+2 = -6k+8 ,  5k+5 = 9k+22k+6k = 8-2 ,  5-2 =9k-5k8k = 6, 4k = 3k = 68, k = 34k = 34 in each case.
Therefore, the point P(2, 5) divides AB in the ratio 3 : 4.

Page No 325:

Question 14:

Let the point P(2, 5) divide AB in the ratio k : 1.
Then, by section formula, the coordinates of P are
x=-6k+8k+1, y=9k+2k+1It is given that the coordinates of P are P(2, 5).2 = -6k+8k+1, 5 =9k+2k+12k+2 = -6k+8 ,  5k+5 = 9k+22k+6k = 8-2 ,  5-2 =9k-5k8k = 6, 4k = 3k = 68, k = 34k = 34 in each case.
Therefore, the point P(2, 5) divides AB in the ratio 3 : 4.

Answer:

Let k : 1 be the ratio in which the point P34, 512 divides the line segment joining the points A12, 32 and 2,-5. Then
34,512=k2+12k+1,k-5+32k+1k2+12k+1=34   and k-5+32k+1=5128k+2=3k+3   and -60k+18=5k+5k=15   and k=15
Hence, the required ratio is 1 : 5.

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Question 15:

Let k : 1 be the ratio in which the point P34, 512 divides the line segment joining the points A12, 32 and 2,-5. Then
34,512=k2+12k+1,k-5+32k+1k2+12k+1=34   and k-5+32k+1=5128k+2=3k+3   and -60k+18=5k+5k=15   and k=15
Hence, the required ratio is 1 : 5.

Answer:

Let the point P(m, 6) divide the line AB in the ratio k : 1.
Then, by the section formula:
x = mx2+nx1m+n, y =my2+ny1m+nThe coordinates of P are (m, 6).m =2k-4k+1 , 6 = 8k+3k+1m(k+1) = 2k-4 ,  6k+6 = 8k+3m(k+1) = 2k-4 ,  6 -3= 8k - 6km(k+1) = 2k-4, 2k =3m(k+1) = 2k-4 , k = 32Therefore, the point P divides the line AB in the ratio 3 : 2.Now, putting the value of k in the equation m(k+1) = 2k-4, we get:m32+1 = 232-4m3+22 = 3-45m2 =-15m = -2m = -25 Therefore, the value of m = -25So, the coordinates of P are (-25, 6).

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Question 16:

Let the point P(m, 6) divide the line AB in the ratio k : 1.
Then, by the section formula:
x = mx2+nx1m+n, y =my2+ny1m+nThe coordinates of P are (m, 6).m =2k-4k+1 , 6 = 8k+3k+1m(k+1) = 2k-4 ,  6k+6 = 8k+3m(k+1) = 2k-4 ,  6 -3= 8k - 6km(k+1) = 2k-4, 2k =3m(k+1) = 2k-4 , k = 32Therefore, the point P divides the line AB in the ratio 3 : 2.Now, putting the value of k in the equation m(k+1) = 2k-4, we get:m32+1 = 232-4m3+22 = 3-45m2 =-15m = -2m = -25 Therefore, the value of m = -25So, the coordinates of P are (-25, 6).

Answer:

Let the point P(−3, k) divide the line AB in the ratio s : 1.
Then, by the section formula:
x = mx2+nx1m+n, y =my2+ny1m+nThe coordinates of P are (-3, k).-3 = -2s-5s+1, k = 3s-4s+1-3s-3 = -2s-5,  ks+1 = 3s-4-3s+2s =-5+3,  ks+1 = 3s-4-s = -2,  ks+1 = 3s-4s =2,  ks+1 = 3s-4Therefore, the point P divides the line AB in the ratio 2 : 1.Now, putting the value of s in the equation ks+1 = 3s-4, we get:k2+1 = 32-43k = 6-43k = 2 k = 23Therefore, the value of k = 23That is, the coordinates of P are (-3, 23).

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Question 17:

Let the point P(−3, k) divide the line AB in the ratio s : 1.
Then, by the section formula:
x = mx2+nx1m+n, y =my2+ny1m+nThe coordinates of P are (-3, k).-3 = -2s-5s+1, k = 3s-4s+1-3s-3 = -2s-5,  ks+1 = 3s-4-3s+2s =-5+3,  ks+1 = 3s-4-s = -2,  ks+1 = 3s-4s =2,  ks+1 = 3s-4Therefore, the point P divides the line AB in the ratio 2 : 1.Now, putting the value of s in the equation ks+1 = 3s-4, we get:k2+1 = 32-43k = 6-43k = 2 k = 23Therefore, the value of k = 23That is, the coordinates of P are (-3, 23).

Answer:

Let AB be divided by the x-axis in the ratio k : 1 at the point P.
Then, by section formula the coordinates of P are
P=5k+2k+1, 6k-3k+1
But P lies on the x-axis; so, its ordinate is 0.
Therefore, 6k-3k+1=06k-3 =06k = 3k =36 k = 12
Therefore, the required ratio is 12 : 1, which is same as 1 : 2.
Thus, the x-axis divides the line AB in the ratio 1 : 2 at the point P.
Applying k = 12, we get the coordinates of point :
  P5k+2k+1, 0 = P5×12+212+1, 0 = P5+421+22, 0 = P93, 0  = P3, 0 
Hence, the point of intersection of AB and the x-axis is P(3, 0).

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Question 18:

Let AB be divided by the x-axis in the ratio k : 1 at the point P.
Then, by section formula the coordinates of P are
P=5k+2k+1, 6k-3k+1
But P lies on the x-axis; so, its ordinate is 0.
Therefore, 6k-3k+1=06k-3 =06k = 3k =36 k = 12
Therefore, the required ratio is 12 : 1, which is same as 1 : 2.
Thus, the x-axis divides the line AB in the ratio 1 : 2 at the point P.
Applying k = 12, we get the coordinates of point :
  P5k+2k+1, 0 = P5×12+212+1, 0 = P5+421+22, 0 = P93, 0  = P3, 0 
Hence, the point of intersection of AB and the x-axis is P(3, 0).

Answer:

Let AB be divided by the x-axis in the ratio k : 1 at the point P.
Then, by section formula the coordinates of P are
P3k-2k+1,7k-3k+1
But P lies on the y-axis; so, its abscissa is 0.
Therefore, 3k-2k+1=03k-2 =03k = 2k =23 k = 23
Therefore, the required ratio is 23 : 1, which is same as 2 : 3.
Thus, the x-axis divides the line AB in the ratio 2:3 at the point P.
Applying k=23, we get the coordinates of point P:
P0,7k-3k+1 = P0,7×23-323+1 = P0,14-932+33 = P0,55  = P0, 1
Hence, the point of intersection of AB and the x-axis is P(0, 1).



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Question 19:

Let AB be divided by the x-axis in the ratio k : 1 at the point P.
Then, by section formula the coordinates of P are
P3k-2k+1,7k-3k+1
But P lies on the y-axis; so, its abscissa is 0.
Therefore, 3k-2k+1=03k-2 =03k = 2k =23 k = 23
Therefore, the required ratio is 23 : 1, which is same as 2 : 3.
Thus, the x-axis divides the line AB in the ratio 2:3 at the point P.
Applying k=23, we get the coordinates of point P:
P0,7k-3k+1 = P0,7×23-323+1 = P0,14-932+33 = P0,55  = P0, 1
Hence, the point of intersection of AB and the x-axis is P(0, 1).

Answer:

Let the line xy − 2 = 0 divide the line segment joining the points A(3, −1) and B(8, 9) in the ratio k : 1 at P.
Then, the coordinates of P are
P8k+3k+1, 9k-1k+1
Since, P lies on the line xy − 2 = 0, we have:

8k+3k+1-9k-1k+1-2=08k+3-9k+1-2k-2=08k-9k-2k+3+1-2=0-3k+2=0-3k=-2k=23
So, the required ratio is 23 : 1, which is equal to 2 : 3.

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Question 20:

Let the line xy − 2 = 0 divide the line segment joining the points A(3, −1) and B(8, 9) in the ratio k : 1 at P.
Then, the coordinates of P are
P8k+3k+1, 9k-1k+1
Since, P lies on the line xy − 2 = 0, we have:

8k+3k+1-9k-1k+1-2=08k+3-9k+1-2k-2=08k-9k-2k+3+1-2=0-3k+2=0-3k=-2k=23
So, the required ratio is 23 : 1, which is equal to 2 : 3.

Answer:

The vertices of ∆ABC are A(0, −1), B(2, 1) and C(0, 3).
Let AD, BE and CF be the medians of ∆ABC.

Let D be the midpoint of BC. So, the coordinates of D are
D2+02,1+32 i.e. D22,42 i.e. D1,2
Let E be the midpoint of AC. So, the coordinates of E are
E0+02,-1+32 i.e. E02,22 i.e. E0,1
Let F be the midpoint of AB. So, the coordinates of F are
F0+22,-1+12  i.e. F22,02  i.e. F1,0
AD = 1-02+2--12 =12+32 = 1+9 = 10 units.BE = 0-22+1-12 =-22+02 = 4+0 = 4 = 2 units.CF = 1-02+0-32 =12+-32 = 1+9 = 10 units.
Therefore, the lengths of the medians: AD10 units, BE = 2 units and CF = 10 units

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Question 21:

The vertices of ∆ABC are A(0, −1), B(2, 1) and C(0, 3).
Let AD, BE and CF be the medians of ∆ABC.

Let D be the midpoint of BC. So, the coordinates of D are
D2+02,1+32 i.e. D22,42 i.e. D1,2
Let E be the midpoint of AC. So, the coordinates of E are
E0+02,-1+32 i.e. E02,22 i.e. E0,1
Let F be the midpoint of AB. So, the coordinates of F are
F0+22,-1+12  i.e. F22,02  i.e. F1,0
AD = 1-02+2--12 =12+32 = 1+9 = 10 units.BE = 0-22+1-12 =-22+02 = 4+0 = 4 = 2 units.CF = 1-02+0-32 =12+-32 = 1+9 = 10 units.
Therefore, the lengths of the medians: AD10 units, BE = 2 units and CF = 10 units

Answer:

Here, (x1 = −1, y1 = 0), (x2 = 5, y2 = −2) and (x3 = 8, y3 = 2).
Let G(x, y) be the centroid of the ∆ABC. Then,
x = 13x1+x2+x3 = 13-1+5+8 = 1312 = 4y = 13y1+y2+y3 = 130-2+2 = 130 = 0
Hence, the centroid of ∆ABC is G(4, 0).

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Question 22:

Here, (x1 = −1, y1 = 0), (x2 = 5, y2 = −2) and (x3 = 8, y3 = 2).
Let G(x, y) be the centroid of the ∆ABC. Then,
x = 13x1+x2+x3 = 13-1+5+8 = 1312 = 4y = 13y1+y2+y3 = 130-2+2 = 130 = 0
Hence, the centroid of ∆ABC is G(4, 0).

Answer:

Two vertices of ∆ABC are A(1, −6) and B(−5, 2). Let the third vertex be C(a, b).
Then the coordinates of its centroid are
C1-5+a3, -6+2+b3C-4+a3, -4+b3
But it is given that G(−2, 1) is the centroid. Therefore,
-2 =-4+a3, 1=-4+b3-6 = -4+a , 3 =-4+b-6+4 = a, 3+4 = ba =-2, b= 7
Therefore, the third vertex of ∆ABC is C(−2, 7).

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Question 23:

Two vertices of ∆ABC are A(1, −6) and B(−5, 2). Let the third vertex be C(a, b).
Then the coordinates of its centroid are
C1-5+a3, -6+2+b3C-4+a3, -4+b3
But it is given that G(−2, 1) is the centroid. Therefore,
-2 =-4+a3, 1=-4+b3-6 = -4+a , 3 =-4+b-6+4 = a, 3+4 = ba =-2, b= 7
Therefore, the third vertex of ∆ABC is C(−2, 7).

Answer:

Two vertices of ∆ABC are B(−3,1) and C(0, −2). Let the third vertex be A(a, b).
Then, the coordinates of its centroid are
  -3+0+a3, 1-2+b3 i.e. -3+a3, -1+b3
But it is given that the centroid is at the origin, that is G(0, 0). Therefore,
0 =-3+a3, 0=-1+b30 = -3+a , 0 =-1+b3 = a, 1= ba =3, b= 1
Therefore, the third vertex of ∆ABC is A(3, 1).

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Question 24:

Two vertices of ∆ABC are B(−3,1) and C(0, −2). Let the third vertex be A(a, b).
Then, the coordinates of its centroid are
  -3+0+a3, 1-2+b3 i.e. -3+a3, -1+b3
But it is given that the centroid is at the origin, that is G(0, 0). Therefore,
0 =-3+a3, 0=-1+b30 = -3+a , 0 =-1+b3 = a, 1= ba =3, b= 1
Therefore, the third vertex of ∆ABC is A(3, 1).

Answer:

The points are A(3, 1), B(0, −2), C(1, 1) and D(4, 4).
Join AC and BD, intersecting at O.   
      

We know that the diagonals of a parallelogram bisect each other.
Midpoint of AC = 3+12,1+12 = 42,22 = 2,1Midpoint of BD = 0+42,-2+42 = 42,22 = 2,1
Thus, the diagonals AC and BD have the same midpoint.
Therefore, ABCD is a parallelogram.

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Question 25:

The points are A(3, 1), B(0, −2), C(1, 1) and D(4, 4).
Join AC and BD, intersecting at O.   
      

We know that the diagonals of a parallelogram bisect each other.
Midpoint of AC = 3+12,1+12 = 42,22 = 2,1Midpoint of BD = 0+42,-2+42 = 42,22 = 2,1
Thus, the diagonals AC and BD have the same midpoint.
Therefore, ABCD is a parallelogram.

Answer:

The points are P(a, −11), Q(5, b), R(2, 15) and S(1, 1).

Join PR and QS, intersecting at O.
We know that the diagonals of a parallelogram bisect each other.
Therefore, O is the midpoint of PR as well as QS.
Midpoint of PR = a+22,-11+152 = a+22,42 = a+22,2Midpoint of QS = 5+12,b+12 = 62,b+12 = 3,b+12 Therefore, a+22=3, b+12=2a+2 = 6 , b+1 = 4a = 6-2 , b = 4-1a = 4 and b = 3

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Question 26:

The points are P(a, −11), Q(5, b), R(2, 15) and S(1, 1).

Join PR and QS, intersecting at O.
We know that the diagonals of a parallelogram bisect each other.
Therefore, O is the midpoint of PR as well as QS.
Midpoint of PR = a+22,-11+152 = a+22,42 = a+22,2Midpoint of QS = 5+12,b+12 = 62,b+12 = 3,b+12 Therefore, a+22=3, b+12=2a+2 = 6 , b+1 = 4a = 6-2 , b = 4-1a = 4 and b = 3

Answer:

Let A(1, −2), B(3, 6) and C(5, 10) be the three vertices of a parallelogram ABCD and the fourth vertex be D(a, b).
Join AC and BD intersecting at O.

We know that the diagonals of a parallelogram bisect each other.
Therefore, O is the midpoint of AC as well as BD.
Midpoint of AC = 1+52,-2+102 = 62,82 = 3,4Midpoint of BD = 3+a2,6+b2 Therefore, 3+a2 = 3 and 6+b2 = 43+a =6 and 6+b =8a = 6-3 and b= 8-6a= 3 and b= 2
Therefore, the fourth vertex is D(3, 2).

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Question 27:

Let A(1, −2), B(3, 6) and C(5, 10) be the three vertices of a parallelogram ABCD and the fourth vertex be D(a, b).
Join AC and BD intersecting at O.

We know that the diagonals of a parallelogram bisect each other.
Therefore, O is the midpoint of AC as well as BD.
Midpoint of AC = 1+52,-2+102 = 62,82 = 3,4Midpoint of BD = 3+a2,6+b2 Therefore, 3+a2 = 3 and 6+b2 = 43+a =6 and 6+b =8a = 6-3 and b= 8-6a= 3 and b= 2
Therefore, the fourth vertex is D(3, 2).

Answer:

Let y-axis divides the line segment joining the points (−4, 7) and (3, −7) in the ratio k : 1. Then
0=3k-4k+13k=4k=43
Hence, the required ratio is 4 : 3.

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Question 28:

Let y-axis divides the line segment joining the points (−4, 7) and (3, −7) in the ratio k : 1. Then
0=3k-4k+13k=4k=43
Hence, the required ratio is 4 : 3.

Answer:

Let the point P12, y divides the line segment joining the points A(3, −5) and B(−7, 9) in the ratio k : 1. Then
12, y=k-7+3k+1, k9-5k+1-7k+3k+1=12 and 9k-5k+1=yk+1=-14k+6k=13
Now, substituting k=13 in 9k-5k+1=y, we get
93-513+1=yy=9-151+3=-32
Hence, required ratio is 1 : 3 and y=-32.

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Question 29:

Let the point P12, y divides the line segment joining the points A(3, −5) and B(−7, 9) in the ratio k : 1. Then
12, y=k-7+3k+1, k9-5k+1-7k+3k+1=12 and 9k-5k+1=yk+1=-14k+6k=13
Now, substituting k=13 in 9k-5k+1=y, we get
93-513+1=yy=9-151+3=-32
Hence, required ratio is 1 : 3 and y=-32.

Answer:

The line segment joining the points A(3, − 3) and B(− 2, 7) is divided by x-axis. Let the required ratio be k : 1. So,
0=k7-3k+1k=37
Now
Point of division=k-2+3k+1, k7-3k+1                         =37×-2+337+1, 37×7-337+1             k=37                         =-6+213+7, 21-213+7                          =32, 0
Hence, the required ratio is 3 : 7 and the point of division is 32, 0.

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Question 30:

The line segment joining the points A(3, − 3) and B(− 2, 7) is divided by x-axis. Let the required ratio be k : 1. So,
0=k7-3k+1k=37
Now
Point of division=k-2+3k+1, k7-3k+1                         =37×-2+337+1, 37×7-337+1             k=37                         =-6+213+7, 21-213+7                          =32, 0
Hence, the required ratio is 3 : 7 and the point of division is 32, 0.

Answer:

Let (x, 0) be the coordinates of R. Then
0=-4+x2x=4
Thus, the coordinates of R are (4, 0).
Here, PQ = QR = PR and the coordinates of P lies on y-axis. Let the coordinates of P be (0, y). Then
PQ=QRPQ2=QR20+42+y-02=82y2=64-16=48y=±43
Hence, the required coordinates are R4, 0 and P0, 43 or P0, -43.

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Question 31:

Let (x, 0) be the coordinates of R. Then
0=-4+x2x=4
Thus, the coordinates of R are (4, 0).
Here, PQ = QR = PR and the coordinates of P lies on y-axis. Let the coordinates of P be (0, y). Then
PQ=QRPQ2=QR20+42+y-02=82y2=64-16=48y=±43
Hence, the required coordinates are R4, 0 and P0, 43 or P0, -43.

Answer:

Let (0, y) be the coordinates of B. Then
0=-3+y2y=3
Thus, the coordinates of B are (0, 3).
Here, AB = BC = AC and by symmetry the coordinates of A lies on x-axis. Let the coordinates of A be (x, 0). Then
AB=BCAB2=BC2x-02+0-32=62x2=36-9=27x=±33
If the coordinates of point A are 33, 0, then the coordinates of D are -33, 0.
If the coordinates of point A are -33, 0, then the coordinates of D are 33, 0.
Hence, the required coordinates are A33, 0, B0, 3 and D-33, 0 or A-33, 0, B0, 3 and D33, 0.

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Question 32:

Let (0, y) be the coordinates of B. Then
0=-3+y2y=3
Thus, the coordinates of B are (0, 3).
Here, AB = BC = AC and by symmetry the coordinates of A lies on x-axis. Let the coordinates of A be (x, 0). Then
AB=BCAB2=BC2x-02+0-32=62x2=36-9=27x=±33
If the coordinates of point A are 33, 0, then the coordinates of D are -33, 0.
If the coordinates of point A are -33, 0, then the coordinates of D are 33, 0.
Hence, the required coordinates are A33, 0, B0, 3 and D-33, 0 or A-33, 0, B0, 3 and D33, 0.

Answer:

Let k be the ratio in which P( −1, y) divides the line segment joining the points A(−3, 10) and B(6, −8). Then
-1, y=k6-3k+1, k-8+10k+1k6-3k+1=-1 and y=k-8+10k+1k=27
Substituting k=27 in  y=k-8+10k+1, we get
y=-8×27+1027+1=-16+709=6
Hence, the required ratio is 2 : 7 and y = 6.

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Question 33:

Let k be the ratio in which P( −1, y) divides the line segment joining the points A(−3, 10) and B(6, −8). Then
-1, y=k6-3k+1, k-8+10k+1k6-3k+1=-1 and y=k-8+10k+1k=27
Substituting k=27 in  y=k-8+10k+1, we get
y=-8×27+1027+1=-16+709=6
Hence, the required ratio is 2 : 7 and y = 6.

Answer:

Here, the points P, Q, R and S are the mid points of AB, BC, CD and DA respectively. Then
Coordinates of P=-1-12, -1+42=-1, 32Coordinates of Q=-1+52, 4+42=2, 4Coordinates of R=5+52, 4-12=5, 32Coordinates of S=-1+52, -1-12=2, -1
Now
PQ=2+12+4-322=9+254=612QR=5-22+32-42=9+254=612RS=5-22+32+12=9+254=612SP=2+12+-1-322=9+254=612PR=5+12+32-322=36=6QS=2-22+-1-42=25=5
Thus, PQ = QR = RS = SP and PRQS therefore PQRS is a rhombus.



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Question 34:

Here, the points P, Q, R and S are the mid points of AB, BC, CD and DA respectively. Then
Coordinates of P=-1-12, -1+42=-1, 32Coordinates of Q=-1+52, 4+42=2, 4Coordinates of R=5+52, 4-12=5, 32Coordinates of S=-1+52, -1-12=2, -1
Now
PQ=2+12+4-322=9+254=612QR=5-22+32-42=9+254=612RS=5-22+32+12=9+254=612SP=2+12+-1-322=9+254=612PR=5+12+32-322=36=6QS=2-22+-1-42=25=5
Thus, PQ = QR = RS = SP and PRQS therefore PQRS is a rhombus.

Answer:

The midpoint of AB is -10-22, 4+02=P-6, 2.
Let k be the ratio in which P divides CD. So
-6, 2=k-4-9k+1, ky-4k+1k-4-9k+1=-6 and ky-4k+1=2k=32
Now, substituting k=32 in ky-4k+1=2, we get
y×32-432+1=23y-85=2y=10+83=6
Hence, the required ratio is 3 : 2 and y = 6.

Page No 327:

Question 35:

The midpoint of AB is -10-22, 4+02=P-6, 2.
Let k be the ratio in which P divides CD. So
-6, 2=k-4-9k+1, ky-4k+1k-4-9k+1=-6 and ky-4k+1=2k=32
Now, substituting k=32 in ky-4k+1=2, we get
y×32-432+1=23y-85=2y=10+83=6
Hence, the required ratio is 3 : 2 and y = 6.

Answer:

Suppose the line intersects the y-axis at P(0, y) and the x-axis at Q(x, 0). 
It is given that (2, –5) is the mid-point of PQ.

Using mid-point formula, we have
x+02,0+y2=2, -5x2, y2=2, -5x2=2 and y2=-5x=4, y=-10
Thus, the coordinates of P and Q are (0, −10) and (4, 0), respectively.

Page No 327:

Question 36:

Suppose the line intersects the y-axis at P(0, y) and the x-axis at Q(x, 0). 
It is given that (2, –5) is the mid-point of PQ.

Using mid-point formula, we have
x+02,0+y2=2, -5x2, y2=2, -5x2=2 and y2=-5x=4, y=-10
Thus, the coordinates of P and Q are (0, −10) and (4, 0), respectively.

Answer:

Let the point P2411, y divides the line PQ in the ratio k : 1.
Then, by the section formula:
x = mx2+nx1m+n, y =my2+ny1m+nThe coordinates of R are 2411, y.2411 =3k+2k+1 , y = 7k-2k+124(k+1)=33k+22 ,  y(k+1)=7k-224k+24=33k+22  ,   yk+y=7k-22=9kk=29
Now consider the equation yk+y =7k-2 and put k=29. 29y+y =149-2 119y =-49y =-411Therefore, the point R divides the line PQ in the ratio 2 : 9.And, the coordinates of R are 2411, -411.
  

Page No 327:

Question 37:

Let the point P2411, y divides the line PQ in the ratio k : 1.
Then, by the section formula:
x = mx2+nx1m+n, y =my2+ny1m+nThe coordinates of R are 2411, y.2411 =3k+2k+1 , y = 7k-2k+124(k+1)=33k+22 ,  y(k+1)=7k-224k+24=33k+22  ,   yk+y=7k-22=9kk=29
Now consider the equation yk+y =7k-2 and put k=29. 29y+y =149-2 119y =-49y =-411Therefore, the point R divides the line PQ in the ratio 2 : 9.And, the coordinates of R are 2411, -411.
  

Answer:


Let the coordinates of A, B, C be (x1,y1), (x2,y2) and (x3,y3) respectively.
Because D is the mid-point of BC, using mid-point formula, we have
x2+x32=3           and          y2+y32=4x2+x3=6        and          y2+y3=8               .....i
Similarly, E is the mid point of AC. Using mid-point formula, we have;
x1+x32=8          and          y1+y32=9x1+x3=16     and          y1+y3=18               .....ii
Again, F is the mid point of AB. Using mid point formula, we have
x1+x22=6          and          y1+y22=7x1+x2=12          and          y1+y2=14               .....iii
Adding (i), (ii) and (iii), we get
2x1+x2+x3=34          and           2y1+y2+y3=40x1+x2+x3=17          and             y1+y2+y3=20               .....iv
On solving equation (iv)using equations (i), (ii) and (iii), we get
x1=11x2=1x3=5
Similarly,
y1=12y2=2y3=6
Hence, the points are: A(11,12), B(1,2) and C(5,6).

Page No 327:

Question 38:


Let the coordinates of A, B, C be (x1,y1), (x2,y2) and (x3,y3) respectively.
Because D is the mid-point of BC, using mid-point formula, we have
x2+x32=3           and          y2+y32=4x2+x3=6        and          y2+y3=8               .....i
Similarly, E is the mid point of AC. Using mid-point formula, we have;
x1+x32=8          and          y1+y32=9x1+x3=16     and          y1+y3=18               .....ii
Again, F is the mid point of AB. Using mid point formula, we have
x1+x22=6          and          y1+y22=7x1+x2=12          and          y1+y2=14               .....iii
Adding (i), (ii) and (iii), we get
2x1+x2+x3=34          and           2y1+y2+y3=40x1+x2+x3=17          and             y1+y2+y3=20               .....iv
On solving equation (iv)using equations (i), (ii) and (iii), we get
x1=11x2=1x3=5
Similarly,
y1=12y2=2y3=6
Hence, the points are: A(11,12), B(1,2) and C(5,6).

Answer:

Let ABCD be the parallelogram with two adjacent vertices A(3, 2) and B(−1, 0). Suppose O(2, −5) be the point of intersection of the diagonals AC and BD.
Let C(x1y1) and D(x2y2) be the coordinates of the other vertices of the parallelogram.

We know that the diagonals of the parallelogram bisect each other. Therefore, O is the mid-point of AC and BD.
Using the mid-point formula, we have
x1+32,y1+22=2,-5x1+32=2 and y1+22=-5x1+3=4 and y1+2=-10x1=4-3=1 and y1=-10-2=-12
So, the coordinates of C are (1, −12).
Also,
x2+-12,y2+02=2,-5x2-12=2 and y22=-5x2-1=4 and y2=-10x2=4+1=5 and y2=-10
So, the coordinates of D are (5, −10).

Page No 327:

Question 39:

Let ABCD be the parallelogram with two adjacent vertices A(3, 2) and B(−1, 0). Suppose O(2, −5) be the point of intersection of the diagonals AC and BD.
Let C(x1y1) and D(x2y2) be the coordinates of the other vertices of the parallelogram.

We know that the diagonals of the parallelogram bisect each other. Therefore, O is the mid-point of AC and BD.
Using the mid-point formula, we have
x1+32,y1+22=2,-5x1+32=2 and y1+22=-5x1+3=4 and y1+2=-10x1=4-3=1 and y1=-10-2=-12
So, the coordinates of C are (1, −12).
Also,
x2+-12,y2+02=2,-5x2-12=2 and y22=-5x2-1=4 and y2=-10x2=4+1=5 and y2=-10
So, the coordinates of D are (5, −10).

Answer:

Given: Point A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD.

Diagonals of a parallelogram bisect each other.

∴ Mid point of AC = Mid point of BD

Mid point of x1, y1 and x2, y2 is x1+x22, y1+y22.Mid point of AC=3+a2, 1+b2Mid point of BD=5+42, 1+32                           =92, 42=92, 23+a2, 1+b2=92, 23+a2=92 and 1+b2=23+a=9 and 1+b=4a=6 and b=3

Hence,  the values of a and is 6 and 3, respectively.

Page No 327:

Question 40:

Given: Point A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD.

Diagonals of a parallelogram bisect each other.

∴ Mid point of AC = Mid point of BD

Mid point of x1, y1 and x2, y2 is x1+x22, y1+y22.Mid point of AC=3+a2, 1+b2Mid point of BD=5+42, 1+32                           =92, 42=92, 23+a2, 1+b2=92, 23+a2=92 and 1+b2=23+a=9 and 1+b=4a=6 and b=3

Hence,  the values of a and is 6 and 3, respectively.

Answer:

Let the points A(2, 1) and B(5, –8) is trisected at the points P(x, y) and Q(a, b).

Thus, AP = PQ = QB

Therefore, P divides AB internally in the ratio 1 : 2. 

Section formula: if the point (xy) divides the line segment joining the points (x1y1) and (x2y2) internally in the ratio : n, then the coordinates (xy) = mx2+nx1m+n, my2+ny1m+n

Therefore, using section formula, the coordinates of P are:

x, y=15+221+2,1-8+211+2x, y=5+43,-8+23x, y=93,-63x, y=3,-2

Hence, the coordinates of P are (3, –2).

Since, P also lies on the line given by 2x – y + k = 0, 
Therefore, (3, –2) satisfies the equation 2x – y + k = 0
23--2+k=06+2+k=0k=-8

Hence,  the values of k is –8.

Page No 327:

Question 41:

Let the points A(2, 1) and B(5, –8) is trisected at the points P(x, y) and Q(a, b).

Thus, AP = PQ = QB

Therefore, P divides AB internally in the ratio 1 : 2. 

Section formula: if the point (xy) divides the line segment joining the points (x1y1) and (x2y2) internally in the ratio : n, then the coordinates (xy) = mx2+nx1m+n, my2+ny1m+n

Therefore, using section formula, the coordinates of P are:

x, y=15+221+2,1-8+211+2x, y=5+43,-8+23x, y=93,-63x, y=3,-2

Hence, the coordinates of P are (3, –2).

Since, P also lies on the line given by 2x – y + k = 0, 
Therefore, (3, –2) satisfies the equation 2x – y + k = 0
23--2+k=06+2+k=0k=-8

Hence,  the values of k is –8.

Answer:

Section formula: if the point (xy) divides the line segment joining the points (x1y1) and (x2y2) internally in the ratio : 1, then the coordinates (xy) = kx2+x1k+1, ky2+y1k+1

Let the point P(0, y) divides the line segment joining the points A(5, –6) and B(–1, 4) in the ratio : 1.

Therefore, using section formula, the coordinates of P are:

0, y=k-1+15k+1,k4+1-6k+10, y=-k+5k+1,4k-6k+10=-k+5k+1 and y=4k-6k+1 0=-k+5k+1-k+5=0k=5Now,  y=4k-6k+1y=45-65+1       k=5y=20-66y=146y=73

Hence, the y-axis divides the line segment joining the points A(5, –6) and B(–1, 4) in the ratio 5 : 1.
and the coordinates of the point of division are 0, 73.



Page No 340:

Question 1:

Section formula: if the point (xy) divides the line segment joining the points (x1y1) and (x2y2) internally in the ratio : 1, then the coordinates (xy) = kx2+x1k+1, ky2+y1k+1

Let the point P(0, y) divides the line segment joining the points A(5, –6) and B(–1, 4) in the ratio : 1.

Therefore, using section formula, the coordinates of P are:

0, y=k-1+15k+1,k4+1-6k+10, y=-k+5k+1,4k-6k+10=-k+5k+1 and y=4k-6k+1 0=-k+5k+1-k+5=0k=5Now,  y=4k-6k+1y=45-65+1       k=5y=20-66y=146y=73

Hence, the y-axis divides the line segment joining the points A(5, –6) and B(–1, 4) in the ratio 5 : 1.
and the coordinates of the point of division are 0, 73.

Answer:

(i) A(1, 2), B(−2, 3) and C(−3, −4) are the vertices of ∆ABC. Then,
(x1 = 1, y1 = 2), (x2 = −2, y2 = 3) and (x3 = −3, y3 = -4)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 1213--4+-2-4-2+-32-3=1213+4-2-6-3-1=127+12+3=1222= 11 sq. units

(ii) A(−5, 7), B(−4, −5) and C(4, 5) are the vertices of ∆ABC. Then,
(x1 = −5, y1 = 7), (x2 = −4, y2 = −5) and (x3 = 4, y3 = 5)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 12-5-5-5+-45-7+47--5=12-5-10-4-2+412=1250+8+48=12106= 53 sq. units

(iii) A(3, 8), B(−4, 2) and C(5, −1) are the vertices of ∆ABC. Then,
(x1 = 3, y1 = 8), (x2 = −4, y2 = 2) and (x3 = 5, y3 = −1)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 1232--1+-4-1-8+58-2=1232+1-4-9+56=129+36+30=1275= 37.5 sq. units

(iv) A(10, −6), B(2, 5) and C(−1, −3) are the vertices of ∆ABC. Then,
(x1 = 10, y1 = −6), (x2 = 2, y2 = 5) and (x3 = −1, y3 = 3)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 12105-3+23--6+-1-6-5=12102+29-1-11=1220+18+11=1249= 24.5 sq. units



Page No 341:

Question 2:

(i) A(1, 2), B(−2, 3) and C(−3, −4) are the vertices of ∆ABC. Then,
(x1 = 1, y1 = 2), (x2 = −2, y2 = 3) and (x3 = −3, y3 = -4)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 1213--4+-2-4-2+-32-3=1213+4-2-6-3-1=127+12+3=1222= 11 sq. units

(ii) A(−5, 7), B(−4, −5) and C(4, 5) are the vertices of ∆ABC. Then,
(x1 = −5, y1 = 7), (x2 = −4, y2 = −5) and (x3 = 4, y3 = 5)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 12-5-5-5+-45-7+47--5=12-5-10-4-2+412=1250+8+48=12106= 53 sq. units

(iii) A(3, 8), B(−4, 2) and C(5, −1) are the vertices of ∆ABC. Then,
(x1 = 3, y1 = 8), (x2 = −4, y2 = 2) and (x3 = 5, y3 = −1)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 1232--1+-4-1-8+58-2=1232+1-4-9+56=129+36+30=1275= 37.5 sq. units

(iv) A(10, −6), B(2, 5) and C(−1, −3) are the vertices of ∆ABC. Then,
(x1 = 10, y1 = −6), (x2 = 2, y2 = 5) and (x3 = −1, y3 = 3)
Area of triangle ABC=12x1y2-y3+x2y3-y1+x3y1-y2= 12105-3+23--6+-1-6-5=12102+29-1-11=1220+18+11=1249= 24.5 sq. units

Answer:

By joining A and C, we get two triangles ABC and ACD.
Let Ax1, y1=A3, -1, Bx2, y2=B9, -5, Cx3, y3=C14, 0 and Dx4, y4=D9, 19. Then
Area of ABC=12x1y2-y3+x2y3-y1+x3y1-y2                       =123-5-0+90+1+14-1+5                       =12-15+9+56=25 sq. units
Area of ACD=12x1y3-y4+x3y4-y1+x4y1-y3                       =1230-19+1419+1+9-1-0                       =12-57+280-9=107 sq. units
So, the area of the quadrilateral is 25 + 107 = 132 sq. units.

Page No 341:

Question 3:

By joining A and C, we get two triangles ABC and ACD.
Let Ax1, y1=A3, -1, Bx2, y2=B9, -5, Cx3, y3=C14, 0 and Dx4, y4=D9, 19. Then
Area of ABC=12x1y2-y3+x2y3-y1+x3y1-y2                       =123-5-0+90+1+14-1+5                       =12-15+9+56=25 sq. units
Area of ACD=12x1y3-y4+x3y4-y1+x4y1-y3                       =1230-19+1419+1+9-1-0                       =12-57+280-9=107 sq. units
So, the area of the quadrilateral is 25 + 107 = 132 sq. units.

Answer:

By joining P and R, we get two triangles PQR and PRS.
Let Px1, y1=P-5, -3, Qx2, y2=Q-4, -6, Rx3, y3=R2, -3 and Sx4, y4=S1, 2. Then
Area of PQR=12x1y2-y3+x2y3-y1+x3y1-y2                       =12-5-6+3-4-3+3+2-3+6                       =1215-0+6=212 sq. units
Area of PRS=12x1y3-y4+x3y4-y1+x4y1-y3                       =12-5-3-2+22+3+1-3+3                       =1225+10+0=352 sq. units
So, the area of the quadrilateral PQRS is 212+352=28 sq. units sq. units.

Page No 341:

Question 4:

By joining P and R, we get two triangles PQR and PRS.
Let Px1, y1=P-5, -3, Qx2, y2=Q-4, -6, Rx3, y3=R2, -3 and Sx4, y4=S1, 2. Then
Area of PQR=12x1y2-y3+x2y3-y1+x3y1-y2                       =12-5-6+3-4-3+3+2-3+6                       =1215-0+6=212 sq. units
Area of PRS=12x1y3-y4+x3y4-y1+x4y1-y3                       =12-5-3-2+22+3+1-3+3                       =1225+10+0=352 sq. units
So, the area of the quadrilateral PQRS is 212+352=28 sq. units sq. units.

Answer:

By joining A and C, we get two triangles ABC and ACD.
Let Ax1, y1=A-3, -1, Bx2, y2=B-2, -4, Cx3, y3=C4, -1 and Dx4, y4=D3, 4. Then
Area of ABC=12x1y2-y3+x2y3-y1+x3y1-y2                       =12-3-4+1-2-1+1+4-1+4                       =129-0+12=212 sq. units
Area of ACD=12x1y3-y4+x3y4-y1+x4y1-y3                       =12-3-1-4+44+1+3-1+1                       =1215+20+0=352 sq. units
So, the area of the quadrilateral ABCD is 212+352=28 sq. units sq. units.

Page No 341:

Question 5:

By joining A and C, we get two triangles ABC and ACD.
Let Ax1, y1=A-3, -1, Bx2, y2=B-2, -4, Cx3, y3=C4, -1 and Dx4, y4=D3, 4. Then
Area of ABC=12x1y2-y3+x2y3-y1+x3y1-y2                       =12-3-4+1-2-1+1+4-1+4                       =129-0+12=212 sq. units
Area of ACD=12x1y3-y4+x3y4-y1+x4y1-y3                       =12-3-1-4+44+1+3-1+1                       =1215+20+0=352 sq. units
So, the area of the quadrilateral ABCD is 212+352=28 sq. units sq. units.

Answer:


Consider the figure. 
Construction: Produce AC by joining points A to C to form two triangles, ABC and ADC
In ABC,
x1=-7, x2=-6 and x3=-3; y1=5, y2=-7 and y3=-8
We know that,
ar(ABC)=12(x1)(y2-y3)+(x2)(y3-y1)+(x3)(y1-y2)ar(ABC)=12(-7)(-7+8)+(-6)(-8-5)+(-3)(5+7)ar(ABC)=12(-7)(1)+(-6)(-13)+(-3)(12)ar(ABC)=12(-7)+78+(-36)ar(ABC)=1235ar(ABC)=352 sq. units
Similarly, in ADC,
x1=-7, x2=2, x3=-3, y1=5, y2=3 and y3=-8
ar(ADC)=12(-7)(3+8)+(2)(-8-5)+(-3)(5-3)ar(ADC)=12(-7)(11)+(2)(-13)+(-3)(2)ar(ADC)=12(-77)-26-6ar(ADC)=12-109ar(ADC)=1092 sq. units
Now, ar(quad. ABCD) = ar(ABC)+ar(ADC)

ar(quad. ABCD) = 352+1092=1442=72
Therefore, area of quadrilateral ABCD is 72 sq. units

Disclaimer: The answer thus calculated does not match with the answer given in the book.
​

Page No 341:

Question 6:


Consider the figure. 
Construction: Produce AC by joining points A to C to form two triangles, ABC and ADC
In ABC,
x1=-7, x2=-6 and x3=-3; y1=5, y2=-7 and y3=-8
We know that,
ar(ABC)=12(x1)(y2-y3)+(x2)(y3-y1)+(x3)(y1-y2)ar(ABC)=12(-7)(-7+8)+(-6)(-8-5)+(-3)(5+7)ar(ABC)=12(-7)(1)+(-6)(-13)+(-3)(12)ar(ABC)=12(-7)+78+(-36)ar(ABC)=1235ar(ABC)=352 sq. units
Similarly, in ADC,
x1=-7, x2=2, x3=-3, y1=5, y2=3 and y3=-8
ar(ADC)=12(-7)(3+8)+(2)(-8-5)+(-3)(5-3)ar(ADC)=12(-7)(11)+(2)(-13)+(-3)(2)ar(ADC)=12(-77)-26-6ar(ADC)=12-109ar(ADC)=1092 sq. units
Now, ar(quad. ABCD) = ar(ABC)+ar(ADC)

ar(quad. ABCD) = 352+1092=1442=72
Therefore, area of quadrilateral ABCD is 72 sq. units

Disclaimer: The answer thus calculated does not match with the answer given in the book.
​

Answer:

The vertices of the triangle are A(2, 1), B(4, 3) and C(2, 5).
Coordinates of midpoint of AB=Px1, y1=2+42,1+32=3, 2Coordinates of midpoint of BC=Qx2, y2=4+22,3+52=3, 4Coordinates of midpoint of AC=Rx3, y3=2+22,1+52=2, 3
Now
Area of PQR=12x1y2-y3+x2y3-y1+x3y1-y2                       =1234-3+33-2+22-4                       =123+3-4=1 sq. unit
Hence, the area of the required triangle is 1 sq. unit.

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Question 7:

The vertices of the triangle are A(2, 1), B(4, 3) and C(2, 5).
Coordinates of midpoint of AB=Px1, y1=2+42,1+32=3, 2Coordinates of midpoint of BC=Qx2, y2=4+22,3+52=3, 4Coordinates of midpoint of AC=Rx3, y3=2+22,1+52=2, 3
Now
Area of PQR=12x1y2-y3+x2y3-y1+x3y1-y2                       =1234-3+33-2+22-4                       =123+3-4=1 sq. unit
Hence, the area of the required triangle is 1 sq. unit.

Answer:

The vertices of the triangle are A(7, −3), B(5, 3), C(3, −1).
Coordinates of D=5+32,3-12=4, 1
For the area of the triangle ADC, let Ax1,y1=A7,-3, Dx2,y2=D4,1 and Cx3,y3=C3,-1. Then
Area of ADC=12x1y2-y3+x2y3-y1+x3y1-y2                       =1271+1+4-1+3+3-3-1                       =1214+8-12=5 sq. unit
Now, for the area of triangle ABD, let Ax1, y1=A7,-3, Bx2, y2=B5, 3 and Dx3, y3=D4, 1. Then
Area of ABD=12x1y2-y3+x2y3-y1+x3y1-y2                       =1273-1+51+3+4-3-3                       =1214+20-24=5 sq. unit
Thus, AreaADC=AreaABD=5 sq. units.
Hence, AD divides ABC into two triangles of equal areas.

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Question 8:

The vertices of the triangle are A(7, −3), B(5, 3), C(3, −1).
Coordinates of D=5+32,3-12=4, 1
For the area of the triangle ADC, let Ax1,y1=A7,-3, Dx2,y2=D4,1 and Cx3,y3=C3,-1. Then
Area of ADC=12x1y2-y3+x2y3-y1+x3y1-y2                       =1271+1+4-1+3+3-3-1                       =1214+8-12=5 sq. unit
Now, for the area of triangle ABD, let Ax1, y1=A7,-3, Bx2, y2=B5, 3 and Dx3, y3=D4, 1. Then
Area of ABD=12x1y2-y3+x2y3-y1+x3y1-y2                       =1273-1+51+3+4-3-3                       =1214+20-24=5 sq. unit
Thus, AreaADC=AreaABD=5 sq. units.
Hence, AD divides ABC into two triangles of equal areas.

Answer:

Let x2, y2 and x3, y3 be the coordinates of B and C respectively. Since, the coordinates of A are (1, −4), therefore
1+x22=2x2=3-4+y22=-1y2=21+x32=0x3=-1-4+y32=-1y3=2
Let Ax1, y1=A1, -4, Bx2, y2=B3, 2 and Cx3, y3=C-1, 2. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                   =1212-2+32+4-1-4-2                   =120+18+6                   =12 sq. units
Hence, the area of the triangle ABC is 12 sq. units.

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Question 9:

Let x2, y2 and x3, y3 be the coordinates of B and C respectively. Since, the coordinates of A are (1, −4), therefore
1+x22=2x2=3-4+y22=-1y2=21+x32=0x3=-1-4+y32=-1y3=2
Let Ax1, y1=A1, -4, Bx2, y2=B3, 2 and Cx3, y3=C-1, 2. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                   =1212-2+32+4-1-4-2                   =120+18+6                   =12 sq. units
Hence, the area of the triangle ABC is 12 sq. units.

Answer:

Let (x, y) be the coordinates of D and x', y' be the coordinates of E. Since, the diagonals of a parallelogram bisect
each other at the same point, therefore
x+82=6+92x=7y+22=1+42y=3
Thus, the coordinates of D are (7, 3).
E is the midpoint of DC, therefore
x'=7+92x'=8y'=3+42y'=72 
Thus, the coordinates of E are 8, 72.
Let Ax1, y1=A6, 1, Ex2, y2=E8, 72 and Dx3, y3=D7, 3. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                   =12672-3+83-1+71-72                   =1232                   =34 sq. unit
Hence, the area of the triangle ADE is 34 sq. units.

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Question 10:

Let (x, y) be the coordinates of D and x', y' be the coordinates of E. Since, the diagonals of a parallelogram bisect
each other at the same point, therefore
x+82=6+92x=7y+22=1+42y=3
Thus, the coordinates of D are (7, 3).
E is the midpoint of DC, therefore
x'=7+92x'=8y'=3+42y'=72 
Thus, the coordinates of E are 8, 72.
Let Ax1, y1=A6, 1, Ex2, y2=E8, 72 and Dx3, y3=D7, 3. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                   =12672-3+83-1+71-72                   =1232                   =34 sq. unit
Hence, the area of the triangle ADE is 34 sq. units.

Answer:

(i) Let Ax1, y1=A1, -3, Bx2, y2=B4, p and Cx3, y3=C-9, 7. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y215=121p-7+47+3-9-3-p15=1210p+6010p+60=30
Therefore
10p+60=-30 or 3010p=-90 or -30p=-9 or -3
Hence, p=-9 or p=-3.

(ii)
Let Ax1, y1=A2, 1, Bx2, y2=B3,-2 and Cx3, y3=C72,y.
Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y25=122-2-y+3y-1+721+210=-4-2y+3y-3+21210=y+7210=y+72    or -10=y+72 y=132 or y = -272
Hence, 132 or -272.
 

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Question 11:

(i) Let Ax1, y1=A1, -3, Bx2, y2=B4, p and Cx3, y3=C-9, 7. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y215=121p-7+47+3-9-3-p15=1210p+6010p+60=30
Therefore
10p+60=-30 or 3010p=-90 or -30p=-9 or -3
Hence, p=-9 or p=-3.

(ii)
Let Ax1, y1=A2, 1, Bx2, y2=B3,-2 and Cx3, y3=C72,y.
Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y25=122-2-y+3y-1+721+210=-4-2y+3y-3+21210=y+7210=y+72    or -10=y+72 y=132 or y = -272
Hence, 132 or -272.
 

Answer:

Let Ax1, y1=Ak+1, 1, Bx2, y2=B4, -3 and Cx3, y3=C7, -k. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y26=12k+1-3+k+4-k-1+71+36=12k2-2k-3-4k-4+28k2-6k+9=0
k-32=0k=3
Hence, k = 3.

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Question 12:

Let Ax1, y1=Ak+1, 1, Bx2, y2=B4, -3 and Cx3, y3=C7, -k. Now
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y26=12k+1-3+k+4-k-1+71+36=12k2-2k-3-4k-4+28k2-6k+9=0
k-32=0k=3
Hence, k = 3.

Answer:

Let Ax1=-2, y1=5, Bx2=k, y2=-4 and Cx3=2k+1, y3=10 be the vertices of
the triangle. So
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y253=12-2-4-10+k10-5+2k+15+453=1228+5k+92k+128+5k+18k+9=106

37+23k=10623k=106-37=69k=6923=3
Hence, k = 3.

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Question 13:

Let Ax1=-2, y1=5, Bx2=k, y2=-4 and Cx3=2k+1, y3=10 be the vertices of
the triangle. So
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y253=12-2-4-10+k10-5+2k+15+453=1228+5k+92k+128+5k+18k+9=106

37+23k=10623k=106-37=69k=6923=3
Hence, k = 3.

Answer:

(i)
Let A(x1 = 2, y1 = −2), B(x2 = −3, y2 = 8) and C(x3 = −1, y3 = 4) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=28-4+-34+2+-1-2-8=8-18+10=0
Hence, the given points are collinear.

(ii)
Let A(x1 = −5, y1 = 1), B(x2 = 5, y2 = 5) and C(x3 = 10, y3 = 7) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=-55-7+57-1+101-5=-5-2+56+10-4=10+30-40=0
Hence, the given points are collinear.

(iii)
Let A(x1 = 5, y1 = 1), B(x2 = 1, y2 = −1) and C(x3 = 11, y3 = 4) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=5-1-4+14-1+111+1=-25+3+22=0
Hence, the given points are collinear.

(iv)
Let A(x1 = 8, y1 = 1), B(x2 = 3, y2 = −4) and C(x3 = 2, y3 = −5) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=8-4+5+3-5-1+21+4=8-18+10=0
Hence, the given points are collinear.



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Question 14:

(i)
Let A(x1 = 2, y1 = −2), B(x2 = −3, y2 = 8) and C(x3 = −1, y3 = 4) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=28-4+-34+2+-1-2-8=8-18+10=0
Hence, the given points are collinear.

(ii)
Let A(x1 = −5, y1 = 1), B(x2 = 5, y2 = 5) and C(x3 = 10, y3 = 7) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=-55-7+57-1+101-5=-5-2+56+10-4=10+30-40=0
Hence, the given points are collinear.

(iii)
Let A(x1 = 5, y1 = 1), B(x2 = 1, y2 = −1) and C(x3 = 11, y3 = 4) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=5-1-4+14-1+111+1=-25+3+22=0
Hence, the given points are collinear.

(iv)
Let A(x1 = 8, y1 = 1), B(x2 = 3, y2 = −4) and C(x3 = 2, y3 = −5) be the given points. Now
x1y2-y3+x2y3-y1+x3y1-y2=8-4+5+3-5-1+21+4=8-18+10=0
Hence, the given points are collinear.

Answer:

Let Ax1, y1=Ax, 2, Bx2, y2=B-3, -4 and Cx3, y3=C7, -5. So, the condition for three collinear points is
x1y2-y3+x2y3-y1+x3y1-y2=0x-4+5-3-5-2+72+4=0x+21+42=0x=-63
Hence, x = − 63.

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Question 15:

Let Ax1, y1=Ax, 2, Bx2, y2=B-3, -4 and Cx3, y3=C7, -5. So, the condition for three collinear points is
x1y2-y3+x2y3-y1+x3y1-y2=0x-4+5-3-5-2+72+4=0x+21+42=0x=-63
Hence, x = − 63.

Answer:

 A(−3, 12), B(7, 6) and C(x, 9) are the given points. Then:
(x1 = −3, y1 = 12), (x2 = 7, y2 = 6) and (x3 = x, y3 = 9)
It is given that the points A, B and C are collinear. Therefore,
x1y2-y3+x2y3-y1+x3y1-y2=0-36-9+79-12+x12-6=0-3-3+7-3+x6=09-21+6x=06x-12=06x = 12x =126 =2
Therefore, when x= 2, the given points are collinear.

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Question 16:

 A(−3, 12), B(7, 6) and C(x, 9) are the given points. Then:
(x1 = −3, y1 = 12), (x2 = 7, y2 = 6) and (x3 = x, y3 = 9)
It is given that the points A, B and C are collinear. Therefore,
x1y2-y3+x2y3-y1+x3y1-y2=0-36-9+79-12+x12-6=0-3-3+7-3+x6=09-21+6x=06x-12=06x = 12x =126 =2
Therefore, when x= 2, the given points are collinear.

Answer:

If the area of the triangle formed by three points is equal to zero, then the points are collinear.

Area of the triangle formed by the vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2.

Now, the given points A(–5, 1), B(1, p) and C(4, –2) are collinear.

Therefore, Area of triangle formed by them is equal to zero.

Area of triangle=012-5p--2+1-2-1+41-p=012-5p+2+1-3+41-p=0-5p-10-3+4-4p=0-9p-9=0-9p-9=0-9p=9p=-1

Hence, the value of p is –1.

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Question 17:

If the area of the triangle formed by three points is equal to zero, then the points are collinear.

Area of the triangle formed by the vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2.

Now, the given points A(–5, 1), B(1, p) and C(4, –2) are collinear.

Therefore, Area of triangle formed by them is equal to zero.

Area of triangle=012-5p--2+1-2-1+41-p=012-5p+2+1-3+41-p=0-5p-10-3+4-4p=0-9p-9=0-9p-9=0-9p=9p=-1

Hence, the value of p is –1.

Answer:

Let A(x1 = −3, y1 = 9), B(x2 = 2, y2 = y) and C(x3 = 4, y3 = −5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=0-3y+5+2-5-9+49-y=0-3y-15-28+36-4y=07y=36-43
y=-1

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Question 18:

Let A(x1 = −3, y1 = 9), B(x2 = 2, y2 = y) and C(x3 = 4, y3 = −5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=0-3y+5+2-5-9+49-y=0-3y-15-28+36-4y=07y=36-43
y=-1

Answer:

Let A(x1 = 8, y1 = 1), B(x2 = 3, y2 = −2k) and C(x3 = ky3 = −5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=08-2k+5+3-5-1+k1+2k=0-16k+40-18+k+2k2=02k2-15k+22=0
2k2-11k-4k+22=0k2k-11-22k-11=0k-22k-11=0k=2 or k=112
Hence, k=2 or k=112.

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Question 19:

Let A(x1 = 8, y1 = 1), B(x2 = 3, y2 = −2k) and C(x3 = ky3 = −5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=08-2k+5+3-5-1+k1+2k=0-16k+40-18+k+2k2=02k2-15k+22=0
2k2-11k-4k+22=0k2k-11-22k-11=0k-22k-11=0k=2 or k=112
Hence, k=2 or k=112.

Answer:

Let A(x1 = 2, y1 = 1), B(x2 = xy2 = y) and C(x3 = 7, y3 = 5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=02y-5+x5-1+71-y=02y-10+4x+7-7y=04x-5y-3=0
Hence, the required relation is 4x − 5y − 3 = 0.

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Question 20:

Let A(x1 = 2, y1 = 1), B(x2 = xy2 = y) and C(x3 = 7, y3 = 5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=02y-5+x5-1+71-y=02y-10+4x+7-7y=04x-5y-3=0
Hence, the required relation is 4x − 5y − 3 = 0.

Answer:

Let A(x1 = xy1 = y), B(x2 = −5, y2 = 7) and C(x3 = −4, y3 = 5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=0x7-5+-55-y+-4y-7=07x-5x-25+5y-4y+28=02x+y+3=0
Hence, the required relation is 2x + y + 3 = 0.

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Question 21:

Let A(x1 = xy1 = y), B(x2 = −5, y2 = 7) and C(x3 = −4, y3 = 5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=0x7-5+-55-y+-4y-7=07x-5x-25+5y-4y+28=02x+y+3=0
Hence, the required relation is 2x + y + 3 = 0.

Answer:

Consider the points A(a, 0), B(0, b) and C(1, 1).
Here, (x1 = a, y1 = 0), (x2 = 0, y2 = b) and (x3 = 1, y3 = 1).
It is given that the points are collinear. So,
x1y2-y3+x2y3-y1+x3y1-y2 = 0ab-1+01-0+10-b=0ab-a-b = oDividing the equation by ab:1-1b-1a = 01-1a+1b = 01a+1b=1 
Therefore, the given points are collinear if 1a+1b = 1.

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Question 22:

Consider the points A(a, 0), B(0, b) and C(1, 1).
Here, (x1 = a, y1 = 0), (x2 = 0, y2 = b) and (x3 = 1, y3 = 1).
It is given that the points are collinear. So,
x1y2-y3+x2y3-y1+x3y1-y2 = 0ab-1+01-0+10-b=0ab-a-b = oDividing the equation by ab:1-1b-1a = 01-1a+1b = 01a+1b=1 
Therefore, the given points are collinear if 1a+1b = 1.

Answer:

Let A(x1 = −3, y1 = 9), B(x2 = ay2 = b) and C(x3 = 4, y3 = 5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=0-3b+5+a-5-9+49-b=0-3b-15-14a+36-4b=02a+b=3
Now, solving a + b = 1 and 2a + b = 3, we get a = 2 and b = −1.
Hence, a = 2 and b = −1.

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Question 23:

Let A(x1 = −3, y1 = 9), B(x2 = ay2 = b) and C(x3 = 4, y3 = 5) be the given points.
The given points are collinear if
x1y2-y3+x2y3-y1+x3y1-y2=0-3b+5+a-5-9+49-b=0-3b-15-14a+36-4b=02a+b=3
Now, solving a + b = 1 and 2a + b = 3, we get a = 2 and b = −1.
Hence, a = 2 and b = −1.

Answer:

Let A(x1 = 0, y1 = −1), B(x2 = 2, y2 = 1) and C(x3 = 0, y3 = 3) be the given points. Then
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                     =1201-3+23+1+0-1-1                     =12×8=4 sq. units
So, the area of the triangle ABC is 4 sq. units.
Let D(a1, b1), E(a2, b2) and F(a3, b3) be the midpoints of AB, BC and AC respectively. Then
a1=0+22=1              b1=-1+12=0a2=2+02=1              b2=1+32=2a3=0+02=0              b3=-1+32=1
Thus, the coordinates of D, E and F are D(a1 = 1, b1 = 0), E(a2 = 1, b2 = 2) and F(a3 = 0, b3 = 1). Now
AreaDEF=12a1b2-b3+a2b3-b1+a3b1-b2                     =1212-1+11-0+00-2                     =121+1+0=1 sq. unit
So, the area of the triangle DEF is 1 sq. unit.
Hence, ABC : DEF=4 : 1.

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Question 24:

Let A(x1 = 0, y1 = −1), B(x2 = 2, y2 = 1) and C(x3 = 0, y3 = 3) be the given points. Then
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                     =1201-3+23+1+0-1-1                     =12×8=4 sq. units
So, the area of the triangle ABC is 4 sq. units.
Let D(a1, b1), E(a2, b2) and F(a3, b3) be the midpoints of AB, BC and AC respectively. Then
a1=0+22=1              b1=-1+12=0a2=2+02=1              b2=1+32=2a3=0+02=0              b3=-1+32=1
Thus, the coordinates of D, E and F are D(a1 = 1, b1 = 0), E(a2 = 1, b2 = 2) and F(a3 = 0, b3 = 1). Now
AreaDEF=12a1b2-b3+a2b3-b1+a3b1-b2                     =1212-1+11-0+00-2                     =121+1+0=1 sq. unit
So, the area of the triangle DEF is 1 sq. unit.
Hence, ABC : DEF=4 : 1.

Answer:

Let A(aa2), B(bb2) and C(0, 0) be the coordinates of the given points.
We know that the area of triangle having vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2 square units.
So,
Area of ∆ABC
=12ab2-0+b0-a2+0a2-b2=12ab2-a2b=12abb-a0             ab0
Since the area of the triangle formed by the points (aa2), (bb2) and (0, 0) is not zero, so the given points are not collinear.

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Question 25:

Let A(aa2), B(bb2) and C(0, 0) be the coordinates of the given points.
We know that the area of triangle having vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2 square units.
So,
Area of ∆ABC
=12ab2-0+b0-a2+0a2-b2=12ab2-a2b=12abb-a0             ab0
Since the area of the triangle formed by the points (aa2), (bb2) and (0, 0) is not zero, so the given points are not collinear.

Answer:

Area of the triangle formed by the vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2.

Now, the given vertices are (x, 3), (4, 4) and  (3, 5)
and the given area is 4 square units.

Therefore,
Area of triangle=12x4-5+45-3+33-44=12x-1+42+3-14=12-x+8-34=12-x+58=-x+5-x+5=8 or -x+5=-8-x=3 or -x=-13x=-3 or x=13

Hence, â€‹the value of x is 13 and −3.



Page No 344:

Question 1:

Area of the triangle formed by the vertices x1, y1, x2, y2 and x3, y3 is 12x1y2-y3+x2y3-y1+x3y1-y2.

Now, the given vertices are (x, 3), (4, 4) and  (3, 5)
and the given area is 4 square units.

Therefore,
Area of triangle=12x4-5+45-3+33-44=12x-1+42+3-14=12-x+8-34=12-x+58=-x+5-x+5=8 or -x+5=-8-x=3 or -x=-13x=-3 or x=13

Hence, â€‹the value of x is 13 and −3.

Answer:

The given points are A(−1, y), B(5, 7) and O(2, −3y).
Here, AO and BO are the radii of the circle. So
AO=BOAO2=BO22+12+-3y-y2=2-52+-3y-729+4y2=-32+3y+729+16y2=9+9y2+49+42y
7y2-42y-49=0y2-6y-7=0y2-7y+y-7=0yy-7+1y-7=0
y-7y+1=0y=-1 or y=7
Hence, y = 7 or y = −1.

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Question 2:

The given points are A(−1, y), B(5, 7) and O(2, −3y).
Here, AO and BO are the radii of the circle. So
AO=BOAO2=BO22+12+-3y-y2=2-52+-3y-729+4y2=-32+3y+729+16y2=9+9y2+49+42y
7y2-42y-49=0y2-6y-7=0y2-7y+y-7=0yy-7+1y-7=0
y-7y+1=0y=-1 or y=7
Hence, y = 7 or y = −1.

Answer:

The given points are A(0, 2), B(3, p) and C(p, 5).
AB=ACAB2=AC23-02+p-22=p-02+5-229+p2-4p+4=p2+94p=4p=1
Hence, p = 1.



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Question 3:

The given points are A(0, 2), B(3, p) and C(p, 5).
AB=ACAB2=AC23-02+p-22=p-02+5-229+p2-4p+4=p2+94p=4p=1
Hence, p = 1.

Answer:

The given vertices are B(4, 0), C(4, 3) and D(0, 3). Here, BD is one of the diagonals. So
BD=4-02+0-32    =42+-32    =16+9    =25    =5
Hence, the length of the diagonal is 5 units..

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Question 4:

The given vertices are B(4, 0), C(4, 3) and D(0, 3). Here, BD is one of the diagonals. So
BD=4-02+0-32    =42+-32    =16+9    =25    =5
Hence, the length of the diagonal is 5 units..

Answer:

The given points are P(k − 1, 2), A(3, k) and B(k, 5).
 AP=BP  AP2=BP2k-1-32+2-k2=k-1-k2+2-52k-42+2-k2=-12+-32
k2-8y+16+4+k2-4k=1+9k2-6y+5=0k-1k-5=0k=1 or k=5
Hence, k = 1 or k = 5.

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Question 5:

The given points are P(k − 1, 2), A(3, k) and B(k, 5).
 AP=BP  AP2=BP2k-1-32+2-k2=k-1-k2+2-52k-42+2-k2=-12+-32
k2-8y+16+4+k2-4k=1+9k2-6y+5=0k-1k-5=0k=1 or k=5
Hence, k = 1 or k = 5.

Answer:

Let k be the ratio in which the point P(x, 2) divides the line joining the points A(x1 = 12, y1 = 5) and B(x2 = 4, y2 = −3). Then
x=k×4+12k+1          and                2=k×-3+5k+1
Now
2=k×-3+5k+12k+2=-3k+5k=35
Hence, the required ratio is 3 : 5.

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Question 6:

Let k be the ratio in which the point P(x, 2) divides the line joining the points A(x1 = 12, y1 = 5) and B(x2 = 4, y2 = −3). Then
x=k×4+12k+1          and                2=k×-3+5k+1
Now
2=k×-3+5k+12k+2=-3k+5k=35
Hence, the required ratio is 3 : 5.

Answer:

The vertices of the rectangle ABCD are A(2, −1), B(5, −1), C(5, 6) and D(2, 6). Now
Coordinates of midpoint of AC=2+52, -1+62=72, 52Coordinates of midpoint of BD=5+22, -1+62=72, 52
Since, the midpoints of AC and BD coincide, therefore the diagonals of rectangle ABCD bisect each other.

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Question 7:

The vertices of the rectangle ABCD are A(2, −1), B(5, −1), C(5, 6) and D(2, 6). Now
Coordinates of midpoint of AC=2+52, -1+62=72, 52Coordinates of midpoint of BD=5+22, -1+62=72, 52
Since, the midpoints of AC and BD coincide, therefore the diagonals of rectangle ABCD bisect each other.

Answer:

The given vertices are A(7, −3), B(5, 3) and C(3, −1).
Since D and E are the midpoints of BC and AC respectively, therefore
Coordinates of D=5+32, 3-12=4, 1Coordinates of E=7+32, -3-12=5, -2
Now
AD=7-42+-3-12=9+16=5BE=5-52+3+22=0+25=5
Hence, AD = BE = 5 units.

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Question 8:

The given vertices are A(7, −3), B(5, 3) and C(3, −1).
Since D and E are the midpoints of BC and AC respectively, therefore
Coordinates of D=5+32, 3-12=4, 1Coordinates of E=7+32, -3-12=5, -2
Now
AD=7-42+-3-12=9+16=5BE=5-52+3+22=0+25=5
Hence, AD = BE = 5 units.

Answer:

Here, the point C(k, 4) divides the join of A(2, 6) and B(5, 1) in the ratio 2 : 3. So
k=2×5+3×22+3  =10+65   =165
Hence, k=165.

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Question 9:

Here, the point C(k, 4) divides the join of A(2, 6) and B(5, 1) in the ratio 2 : 3. So
k=2×5+3×22+3  =10+65   =165
Hence, k=165.

Answer:

Let P(x, 0) be the point on x-axis. Then
AP=BPAP2=BP2x+12+0-02=x-52+0-02x2+2x+1=x2-10x+2512x=24x=2
Hence, x = 2.

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Question 10:

Let P(x, 0) be the point on x-axis. Then
AP=BPAP2=BP2x+12+0-02=x-52+0-02x2+2x+1=x2-10x+2512x=24x=2
Hence, x = 2.

Answer:

The given points are A-85,2 and B25,2.
Then, x1=-85, y1=2 and x2=25, y2=2
Therefore,
AB=x2-x12+y2- y12    =25--852+2-22    =22+02    =4+0    =4    =2 units

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Question 11:

The given points are A-85,2 and B25,2.
Then, x1=-85, y1=2 and x2=25, y2=2
Therefore,
AB=x2-x12+y2- y12    =25--852+2-22    =22+02    =4+0    =4    =2 units

Answer:

The point  3, a lies on the line 2x-3y=5.If point 3, a lies on the line 2x-3y=5 , then2x-3y=52×3-3×a=5
              6-3a=53a=1a=13
Hence, the value of a is 13.

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Question 12:

The point  3, a lies on the line 2x-3y=5.If point 3, a lies on the line 2x-3y=5 , then2x-3y=52×3-3×a=5
              6-3a=53a=1a=13
Hence, the value of a is 13.

Answer:

The given points  A4, 3 and Bx, 5 lie on the circle with centre O2,3.Then, OA = OB x-22+5-32=4-22+3-32x-22+22=22+02x-22=22-22x-22=0x-2=0x=2
Hence, the value of x=2.

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Question 13:

The given points  A4, 3 and Bx, 5 lie on the circle with centre O2,3.Then, OA = OB x-22+5-32=4-22+3-32x-22+22=22+02x-22=22-22x-22=0x-2=0x=2
Hence, the value of x=2.

Answer:

Let the point Px, y be equidistant from the points A(7, 1) and B(3, 5).
Then,
PA=PBPA2=PB2x-72+y-12=x-32+y-52x2+y2-14x-2y+50=x2+y2-6x-10y+348x-8y=16x-y=2

Page No 345:

Question 14:

Let the point Px, y be equidistant from the points A(7, 1) and B(3, 5).
Then,
PA=PBPA2=PB2x-72+y-12=x-32+y-52x2+y2-14x-2y+50=x2+y2-6x-10y+348x-8y=16x-y=2

Answer:

The given points are A(a, b), B(b, c) and C(c, a).
Here,
 x1=a, y1=b, x2=b, y2=c and x3=c, y3=a
Let the centroid be (x, y).
Then,
x=13x1+x2+x3  =13a+b+c  =a+b+c3y=13 y1+ y2+ y3  =13b+c+a  =a+b+c3
But it is given that the centroid of the triangle is the origin.
Then, we have:
a+b+c3=0a+b+c=0

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Question 15:

The given points are A(a, b), B(b, c) and C(c, a).
Here,
 x1=a, y1=b, x2=b, y2=c and x3=c, y3=a
Let the centroid be (x, y).
Then,
x=13x1+x2+x3  =13a+b+c  =a+b+c3y=13 y1+ y2+ y3  =13b+c+a  =a+b+c3
But it is given that the centroid of the triangle is the origin.
Then, we have:
a+b+c3=0a+b+c=0

Answer:

The given points are A(2, 2), B(−4, −4) and C(5, −8).
Here, x1=2, y1=2, x2=-4, y2=-4 and x3=5, y3=-8
Let G(x, y) be the centroid of ABC. Then,
x=13x1+x2+x3  =132-4+5  =1
y=13y1+y2+y3  =132-4-8  =-103
Hence, the centroid of  ABC is G1,-103.

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Question 16:

The given points are A(2, 2), B(−4, −4) and C(5, −8).
Here, x1=2, y1=2, x2=-4, y2=-4 and x3=5, y3=-8
Let G(x, y) be the centroid of ABC. Then,
x=13x1+x2+x3  =132-4+5  =1
y=13y1+y2+y3  =132-4-8  =-103
Hence, the centroid of  ABC is G1,-103.

Answer:

Let the required ratio be k : 1.
Then, by section formula, the coordinates of C are
C7k+2k+1,8k+3k+1
Therefore,
7k+2k+1=4  and  8k+3k+1=5                 C4, 5 is given7k+2=4k+4  and  8k+3=5k+53k=2  and  3k=2
k=23in each case
So, the required ratio is 23:1, which is same as 2:3.

Page No 345:

Question 17:

Let the required ratio be k : 1.
Then, by section formula, the coordinates of C are
C7k+2k+1,8k+3k+1
Therefore,
7k+2k+1=4  and  8k+3k+1=5                 C4, 5 is given7k+2=4k+4  and  8k+3=5k+53k=2  and  3k=2
k=23in each case
So, the required ratio is 23:1, which is same as 2:3.

Answer:

The given points are A2, 3, B4, k and C6, -3.
Here, x1=2, y1=3, x2=4, y2=k and x3=6, y3=-3
It is given that the points A, B and C​ are collinear. Then,
x1y2-y3+x2y3-y1+x3y1-y2=02k+3+4-3-3+63-k=02k+6-24+18-6k=0-4k=0k=0



Page No 348:

Question 1:

The given points are A2, 3, B4, k and C6, -3.
Here, x1=2, y1=3, x2=4, y2=k and x3=6, y3=-3
It is given that the points A, B and C​ are collinear. Then,
x1y2-y3+x2y3-y1+x3y1-y2=02k+3+4-3-3+63-k=02k+6-24+18-6k=0-4k=0k=0

Answer:

The distance of a point (x, y) from the origin O(0, 0) is x2+y2.
Let P(x = −6, y = 8) be the given point. Then
OP=x2+y2     =-62+82     =36+64     =100=10
Hence, the correct answer is option (d).

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Question 2:

The distance of a point (x, y) from the origin O(0, 0) is x2+y2.
Let P(x = −6, y = 8) be the given point. Then
OP=x2+y2     =-62+82     =36+64     =100=10
Hence, the correct answer is option (d).

Answer:

The distance of a point (x, y) from x-axis is y.
Here, the point is (−3, 4). So, its distance from x-axis is 4=4.
Hence, the correct answer is option (c).

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Question 3:

The distance of a point (x, y) from x-axis is y.
Here, the point is (−3, 4). So, its distance from x-axis is 4=4.
Hence, the correct answer is option (c).

Answer:

Let P(x, 0) the point on x-axis, then
AP=BPAP2=BP2x+12+0-02=x-52+0-02x2+2x+1=x2-10x+2512x=24x=2
Thus, the required point is (2, 0).
Hence, the correct answer is option (b).

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Question 4:

Let P(x, 0) the point on x-axis, then
AP=BPAP2=BP2x+12+0-02=x-52+0-02x2+2x+1=x2-10x+2512x=24x=2
Thus, the required point is (2, 0).
Hence, the correct answer is option (b).

Answer:

Since R(5, 6) is the midpoint of the line segment AB joining the points A(6, 5) and B(4, y), therefore
5+y2=65+y=12y=12-5=7
Hence, the correct option is (b).

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Question 5:

Since R(5, 6) is the midpoint of the line segment AB joining the points A(6, 5) and B(4, y), therefore
5+y2=65+y=12y=12-5=7
Hence, the correct option is (b).

Answer:

The point C(k, 4) divides the join of the points A(2, 6) and B(5, 1) in the ratio 2 : 3. So
k=2×5+3×22+3=10+65=165
Hence, the correct answer is option (c).

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Question 6:

The point C(k, 4) divides the join of the points A(2, 6) and B(5, 1) in the ratio 2 : 3. So
k=2×5+3×22+3=10+65=165
Hence, the correct answer is option (c).

Answer:

Let A(0, 4), B(0, 0) and C(3, 0) be the given vertices. So
AB=0-02+4-02=16=4BC=0-32+0-02=9=3AC=0-32+4-02=9+16=5
Therefore
AB + BC + AC = 4 + 3 + 5 = 12
Hence, the correct answer is option (d).

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Question 7:

Let A(0, 4), B(0, 0) and C(3, 0) be the given vertices. So
AB=0-02+4-02=16=4BC=0-32+0-02=9=3AC=0-32+4-02=9+16=5
Therefore
AB + BC + AC = 4 + 3 + 5 = 12
Hence, the correct answer is option (d).

Answer:

The diagonals of a parallelogram bisect each other. The vertices of the ||gm ABCD are A(1, 3), B(−1, 2) and C(2, 5) and D(x, 4).
Here, AC and BD are the diagonals. So
1+22=-1+x2x-1=3x=1+3=4
Hence, the correct answer is option (b).

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Question 8:

The diagonals of a parallelogram bisect each other. The vertices of the ||gm ABCD are A(1, 3), B(−1, 2) and C(2, 5) and D(x, 4).
Here, AC and BD are the diagonals. So
1+22=-1+x2x-1=3x=1+3=4
Hence, the correct answer is option (b).

Answer:

Let A(x1 = x, y1 = 2), B(x2 = −3, y2 = −4) and C(x3 = 7, y3 = −5) be collinear points. Then
x1y2-y3+x2y3-y1+x3y1-y2=0x-4+5+-3-5-2+72+4=0x+21+42=0x=-63
Hence, the correct answer is option (a).

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Question 9:

Let A(x1 = x, y1 = 2), B(x2 = −3, y2 = −4) and C(x3 = 7, y3 = −5) be collinear points. Then
x1y2-y3+x2y3-y1+x3y1-y2=0x-4+5+-3-5-2+72+4=0x+21+42=0x=-63
Hence, the correct answer is option (a).

Answer:

Let A(x1 = 5, y1 = 0), B(x2 = 8, y2 = 0) and C(x3 = 8, y3 = 4) be the vertices of the triangle. Then
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                     =1250-4+84-0+80-0                     =12-20+32+0                     =6 sq. units
Hence, the correct answer is option (c).

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Question 10:

Let A(x1 = 5, y1 = 0), B(x2 = 8, y2 = 0) and C(x3 = 8, y3 = 4) be the vertices of the triangle. Then
AreaABC=12x1y2-y3+x2y3-y1+x3y1-y2                     =1250-4+84-0+80-0                     =12-20+32+0                     =6 sq. units
Hence, the correct answer is option (c).

Answer:

Let A(x1 = a, y1 = 0), O(x2 = 0, y2 = 0) and B(x3 = 0, y3 = b) be the given vertices. So
AreaABO=12x1y2-y3+x2y3-y1+x3y1-y2                    =12a0-b+0b-0+00-0                    =12-ab=                    =12ab
Hence, the correct answer is (b).

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Question 11:

Let A(x1 = a, y1 = 0), O(x2 = 0, y2 = 0) and B(x3 = 0, y3 = b) be the given vertices. So
AreaABO=12x1y2-y3+x2y3-y1+x3y1-y2                    =12a0-b+0b-0+00-0                    =12-ab=                    =12ab
Hence, the correct answer is (b).

Answer:

The point Pa2, 4 is the midpoint of the line segment joining the points A(−6, 5) and B(−2, 3). So
a2=-6-22a2=-4a=-8
Hence, the correct answer is option (a).



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Question 12:

The point Pa2, 4 is the midpoint of the line segment joining the points A(−6, 5) and B(−2, 3). So
a2=-6-22a2=-4a=-8
Hence, the correct answer is option (a).

Answer:

Here, AC and BD are two diagonals of the rectangle ABCD. So
BD=4-02+0-32    =42+-32    =16+9    =25    =5 units
Hence, the correct answer is option (a).

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Question 13:

Here, AC and BD are two diagonals of the rectangle ABCD. So
BD=4-02+0-32    =42+-32    =16+9    =25    =5 units
Hence, the correct answer is option (a).

Answer:

Here, the point P divides the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2 : 1. Then
Coordinates of P=2×4+1×12+1, 2×6+1×32+1                           =8+13, 12+33                           =93, 153                           =3, 5
Hence, the correct answer is option (b).

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Question 14:

Here, the point P divides the line segment joining the points A(1, 3) and B(4, 6) in the ratio 2 : 1. Then
Coordinates of P=2×4+1×12+1, 2×6+1×32+1                           =8+13, 12+33                           =93, 153                           =3, 5
Hence, the correct answer is option (b).

Answer:

Let (x, y) be the coordinates of the other end of the diameter. Then
-2=2+x2x=-65=3+y2y=7
Hence, the correct answer is option (a).

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Question 15:

Let (x, y) be the coordinates of the other end of the diameter. Then
-2=2+x2x=-65=3+y2y=7
Hence, the correct answer is option (a).

Answer:

Here, AQ : BQ = 2 : 1. Then
y=2×-5+1×-22+1  =-10-23  =-4
Hence, the correct answer is option (c).

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Question 16:

Here, AQ : BQ = 2 : 1. Then
y=2×-5+1×-22+1  =-10-23  =-4
Hence, the correct answer is option (c).

Answer:

Let (x, y) be the coordinates of A. Then
0=-2+x2x=24=3+y2y=8-3=5
Thus, the coordinates of A are (2, 5).
Hence, the correct answer is option (a).

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Question 17:

Let (x, y) be the coordinates of A. Then
0=-2+x2x=24=3+y2y=8-3=5
Thus, the coordinates of A are (2, 5).
Hence, the correct answer is option (a).

Answer:

Let (x, y) be the coordinates of P. Then
x=2×5+3×22+3=10+65=165y=2×2+3×-52+3=4-155=-115
Thus, the coordinates of point P are 165, -115 and so it lies in the fourth quadrant.
Hence, the correct answer is option (d).

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Question 18:

Let (x, y) be the coordinates of P. Then
x=2×5+3×22+3=10+65=165y=2×2+3×-52+3=4-155=-115
Thus, the coordinates of point P are 165, -115 and so it lies in the fourth quadrant.
Hence, the correct answer is option (d).

Answer:

The given points are A(−6, 7) and B(−1, −5). So
AB=-6+12+7+52    =-52+122    =25+144    =169    =13
Thus, 2AB = 26.
Hence, the correct answer is option (b).

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Question 19:

The given points are A(−6, 7) and B(−1, −5). So
AB=-6+12+7+52    =-52+122    =25+144    =169    =13
Thus, 2AB = 26.
Hence, the correct answer is option (b).

Answer:

Let P(x, 0) be the point on x-axis. Then as per the question
AP=BPAP2=BP2x-72+0-62=x+32+0-42x2-14x+49+36=x2+6x+9+1660=20xx=6020=3
Thus, the required point is (3, 0).
Hence, the correct answer is option (c).

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Question 20:

Let P(x, 0) be the point on x-axis. Then as per the question
AP=BPAP2=BP2x-72+0-62=x+32+0-42x2-14x+49+36=x2+6x+9+1660=20xx=6020=3
Thus, the required point is (3, 0).
Hence, the correct answer is option (c).

Answer:

(b) 4 units
The y-coordinate is the distance of the point from the x-axis.
Here, the y-coordinate is 4.

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Question 21:

(b) 4 units
The y-coordinate is the distance of the point from the x-axis.
Here, the y-coordinate is 4.

Answer:

(c) 1 : 2
Let AB be divided by the x axis in the ratio k : 1 at the point P.
Then, by section formula, the coordinates of are
P5k+2k+1,6k-3k+1
Butlies on the x axis: so, its ordinate is 0.
6k-3k+1=0
6k-3=0
6k=3
k=12
Hence, the required ratio is 12 : 1, which is same as 1 : 2.

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Question 22:

(c) 1 : 2
Let AB be divided by the x axis in the ratio k : 1 at the point P.
Then, by section formula, the coordinates of are
P5k+2k+1,6k-3k+1
Butlies on the x axis: so, its ordinate is 0.
6k-3k+1=0
6k-3=0
6k=3
k=12
Hence, the required ratio is 12 : 1, which is same as 1 : 2.

Answer:

(d) 1 : 2
Let AB be divided by the y axis in the ratio k : 1 at the point P.
Then, by section formula, the coordinates of are
P8k-4k+1,3k+2k+1
But, P lies on the y axis; so, its abscissa is 0.
8k-4k+1=0
8k-4=0
8k=4
k=12
Hence, the required ratio is 12 : 1, which is same as 1 : 2.

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Question 23:

(d) 1 : 2
Let AB be divided by the y axis in the ratio k : 1 at the point P.
Then, by section formula, the coordinates of are
P8k-4k+1,3k+2k+1
But, P lies on the y axis; so, its abscissa is 0.
8k-4k+1=0
8k-4=0
8k=4
k=12
Hence, the required ratio is 12 : 1, which is same as 1 : 2.

Answer:

(b) −1
The given points are A(−3, b) and B(1, b+4).
Then, x1=-3, y1=b and x2=1, y2=b+4
Therefore,
x=-3+12  =-22  =-1
and
y=b+b+42  =2b+42  =b+2
But the midpoint is P-1, 1.
Therefore,
b+2=1b=-1



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Question 24:

(b) −1
The given points are A(−3, b) and B(1, b+4).
Then, x1=-3, y1=b and x2=1, y2=b+4
Therefore,
x=-3+12  =-22  =-1
and
y=b+b+42  =2b+42  =b+2
But the midpoint is P-1, 1.
Therefore,
b+2=1b=-1

Answer:

(b) 2 : 9
Let the line​ 2x+y-4=0 divide the line segment in the ratio k : 1 at the point P.
Then, by section formula, the coordinates of P are
P3k+2k+1,7k-2k+1
Since P lies on the line 2x+y-4=0 , we have:
23k+2k+1+7k-2k+1-4=06k+4+7k-2-4k+4=09k=2k=29
Hence, the required ratio is 29 : 1, which is same as 2 : 9.

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Question 25:

(b) 2 : 9
Let the line​ 2x+y-4=0 divide the line segment in the ratio k : 1 at the point P.
Then, by section formula, the coordinates of P are
P3k+2k+1,7k-2k+1
Since P lies on the line 2x+y-4=0 , we have:
23k+2k+1+7k-2k+1-4=06k+4+7k-2-4k+4=09k=2k=29
Hence, the required ratio is 29 : 1, which is same as 2 : 9.

Answer:

(c) 72,92
D is the midpoint of BC.
So, the coordinates of D are
D6+12,5+42    B6, 5 and C1, 4x1=6, y1=5 and x2=1, y2=4i.e. D72, 92

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Question 26:

(c) 72,92
D is the midpoint of BC.
So, the coordinates of D are
D6+12,5+42    B6, 5 and C1, 4x1=6, y1=5 and x2=1, y2=4i.e. D72, 92

Answer:

(d) (4, 0)
The given points are A-1, 0, B5, -2 and C8, 2.
Here, x1=-1, y1=0, x2=5, y2=-2 and x3=8, y3=2
Let G(x, y) be the centroid of ABC. Then,
x=13x1+x2+x3  =13-1+5+8  =4
and
y=13y1+y2+y3  =130-2+2  =0
Hence, the centroid of ABC is G(4, 0).

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Question 27:

(d) (4, 0)
The given points are A-1, 0, B5, -2 and C8, 2.
Here, x1=-1, y1=0, x2=5, y2=-2 and x3=8, y3=2
Let G(x, y) be the centroid of ABC. Then,
x=13x1+x2+x3  =13-1+5+8  =4
and
y=13y1+y2+y3  =130-2+2  =0
Hence, the centroid of ABC is G(4, 0).

Answer:

(c) (−4, −15)
Two vertices of ABC are A-1,4 and B5,2.
Let the third vertex be C(a, b).
Then, the coordinates of its centroid are
G-1+5+a3,4+2+b3i.e. G4+a3,6+b3
But it is given that the centroid is G0,-3.
Therefore,
4+a3=0 and 6+b3=-3
4+a=0 and 6+b=-9
a=-4 and b=-15
Hence, the third vertex of ABC is C-4, -15.

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Question 28:

(c) (−4, −15)
Two vertices of ABC are A-1,4 and B5,2.
Let the third vertex be C(a, b).
Then, the coordinates of its centroid are
G-1+5+a3,4+2+b3i.e. G4+a3,6+b3
But it is given that the centroid is G0,-3.
Therefore,
4+a3=0 and 6+b3=-3
4+a=0 and 6+b=-9
a=-4 and b=-15
Hence, the third vertex of ABC is C-4, -15.

Answer:

(a) isosceles
Let A(−4, 0), B(4, 0) and C(0, 3) be the given points. Then,
AB=4+42+0-02    =82+02    =64+0    =64    =8 unitsBC=0-42+3-02    =-42+32    =16+9    =25    =5 unitsAC=0+42+3-02    =42+32    =16+9    =25    =5 units
BC = AC = 5 units
Therefore, ABC is isosceles.

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Question 29:

(a) isosceles
Let A(−4, 0), B(4, 0) and C(0, 3) be the given points. Then,
AB=4+42+0-02    =82+02    =64+0    =64    =8 unitsBC=0-42+3-02    =-42+32    =16+9    =25    =5 unitsAC=0+42+3-02    =42+32    =16+9    =25    =5 units
BC = AC = 5 units
Therefore, ABC is isosceles.

Answer:

(d) right-angled

Let P(0, 6), Q(−5, 3) and R(3, 1) be the given points. Then,
PQ=-5-02+3-62    =-52+-32    =25+9    =34  unitsQR=3+52+1-32    =82+-22    =64+4    =68    =217 unitsPR=3-02+1-62    =32+-52    =9+25    =34 unitsPQ2+PR2342+342=68QR2217 2=68
Thus, PQ2+PR2=QR2
Therefore, ∆PQR is right-angled.

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Question 30:

(d) right-angled

Let P(0, 6), Q(−5, 3) and R(3, 1) be the given points. Then,
PQ=-5-02+3-62    =-52+-32    =25+9    =34  unitsQR=3+52+1-32    =82+-22    =64+4    =68    =217 unitsPR=3-02+1-62    =32+-52    =9+25    =34 unitsPQ2+PR2342+342=68QR2217 2=68
Thus, PQ2+PR2=QR2
Therefore, ∆PQR is right-angled.

Answer:

(b) k = 6
The given points are A(2, 3), B(5, k) and C(6, 7).
Here, x1=2, y1=3, x2=5, y2=k and x3=6, y3=7.
Points A,B and C are collinear. Then,
x1y2-y3+x2y3-y1+x3y1-y2=02k-7+57-3+63-k=02k-14+20+18-6k=0-4k=-24k=6

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Question 31:

(b) k = 6
The given points are A(2, 3), B(5, k) and C(6, 7).
Here, x1=2, y1=3, x2=5, y2=k and x3=6, y3=7.
Points A,B and C are collinear. Then,
x1y2-y3+x2y3-y1+x3y1-y2=02k-7+57-3+63-k=02k-14+20+18-6k=0-4k=-24k=6

Answer:

(c) 2a = b
The given points are A(1, 2), O(0, 0) and C(a, b).
Here, x1=1, y1=2, x2=0, y2=0 and x3=a, y3=b.
Points A, O and C are collinear.
x1y2-y3+x2y3-y1+x3y1-y2=010-b+0b-2+a2-0=0-b+2a=02a=b

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Question 32:

(c) 2a = b
The given points are A(1, 2), O(0, 0) and C(a, b).
Here, x1=1, y1=2, x2=0, y2=0 and x3=a, y3=b.
Points A, O and C are collinear.
x1y2-y3+x2y3-y1+x3y1-y2=010-b+0b-2+a2-0=0-b+2a=02a=b

Answer:

(c) 8 sq units
The given points are A3, 0, B7, 0 and C8, 4.
Here, x1=3, y1=0, x2=7, y2=0 and x3=8, y3=4
Therefore,
Area of ABC=12x1y2-y3+x2y3-y1+x3y1-y2

                  =1230-4+74-0+80-0=12-12+28+0=12×16=8 sq units

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Question 33:

(c) 8 sq units
The given points are A3, 0, B7, 0 and C8, 4.
Here, x1=3, y1=0, x2=7, y2=0 and x3=8, y3=4
Therefore,
Area of ABC=12x1y2-y3+x2y3-y1+x3y1-y2

                  =1230-4+74-0+80-0=12-12+28+0=12×16=8 sq units

Answer:

(c) 34 units
A0,3, O0,0 and B5,0 are the three vertices of a rectangle; let C be the fourth vertex.
Then, the length of the diagonal,
AB=5-02+0-32    =52+-32    =25+9    =34 units
Since, the diagonals of rectangle are equal .
Hence, the length of its diagonals is 34 units.

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Question 34:

(c) 34 units
A0,3, O0,0 and B5,0 are the three vertices of a rectangle; let C be the fourth vertex.
Then, the length of the diagonal,
AB=5-02+0-32    =52+-32    =25+9    =34 units
Since, the diagonals of rectangle are equal .
Hence, the length of its diagonals is 34 units.

Answer:

(c) p = ±4 only
The given points are A(4, p) and B(1, 0) and AB = 5.
Then, x1=4, y1=p and x2=1, y2=0
Therefore,
AB=5x2-x12+y2-y12=51-42+0-p2=5-32+-p2=259+p2=25p2=16p=±16p=±4



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