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In what ratio the line segment joining the points (-2,-3) and (3,7) is divided by the y – axis ? Also, find the co – ordinates of the point of division

Let y-axis intersect the line segment joining the points A (-2,-3) and B(3,7) in the ratio at a point P(0,y).

So apply section formula in order to get,

Equate the x component to get,

Using the value of lamda we can get the co-ordinate of P as-

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eq. of line joining the 2 points =>(y+3)(x-3)=(y-7)(x+2) =>xy-3y+3x-9=xy+2y-7x-14 =>10x-5y+5=o =>2x-y+2=0. Now, we find the point where this line intersects the y-axis.....i.e we put x=0 . So, we have that the point is (0,2). Now, applying section formula, we find the ratio in which (0,2) divides  the line segment joining (-2,-3)&(3,7). Let the ratio be k:1.0=(3k-2)/k+1 =>k=2/3. Therefore, the ratio is 2:3 and the coordinates of the point of division is (0,2).

 

 

                                                       

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Let the required point be P( 0, y)    [B'coz the point lies on y-axis]
Also, let the required ratio be k : 1

Using section formula,
P(0,y) = [(m1*x2+m2*x1) / m1+m2 , (m1*y2 +m2*y1) / m1+ m2]

Substituting values m1=k , m2=1, x1= -2 ,x2= 3 ,  y1=-3, y2=7, We get
P(0.y) = [ (3*k +1*-2)/ k+1 , (7*k +1*-3) /k+1]

P(0,y) = [ (3k-2)/ k+1 , (7k-3)/ k+1]
0= (3k-2)/k+1...(a)  ,  y= (7k-3)/k+1......(b)

(a) ..3k-2 = 0 

3k = 2

k=2/3

Sub k= 2/3 in (b)

y = (7*(2/3) - 3)/ (2/3)+1

   =(14/3 - 3)/ 5/3

   = (14-9)/3 / (5/3)

   = (5/3) /(5/3)

   = 1

Therefore ratio is k :1 = 2/3 :1 = 2:3
and point of division is P( 0,1)

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