Rd Sharma Xi 2018 Solutions for Class 12 Science Math Chapter 26 Ellipse are provided here with simple step-by-step explanations. These solutions for Ellipse are extremely popular among Class 12 Science students for Math Ellipse Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2018 Book of Class 12 Science Math Chapter 26 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2018 Solutions. All Rd Sharma Xi 2018 Solutions for class Class 12 Science Math are prepared by experts and are 100% accurate.

#### Question 1:

Find the equation of the ellipse whose focus is (1, −2), the directrix 3x − 2y + 5 = 0 and eccentricity equal to 1/2.

#### Question 2:

Find the equation of the ellipse in the following cases:
(i) focus is (0, 1), directrix is x + y = 0 and e = $\frac{1}{2}$
(ii) focus is (−1, 1), directrix is xy + 3 = 0 and e = $\frac{1}{2}$
(iii) focus is (−2, 3), directrix is 2x + 3y + 4 = 0 and e = $\frac{4}{5}$
(iv) focus is (1, 2), directrix is 3x + 4y − 5 = 0 and e = $\frac{1}{2}$.

(i)

(ii)

(iii)

(iv)

#### Question 3:

Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
(i) 4x2 + 9y2 = 1
(ii) 5x2 + 4y2 = 1
(iii) 4x2 + 3y2 = 1
(iv) 25x2 + 16y2 = 1600.
(v) 9x2 + 25y2 = 225

#### Question 4:

Find the equation to the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point (−3, 1) and has eccentricity $\sqrt{\frac{2}{5}}$.

#### Question 5:

Find the equation of the ellipse in the following cases:
(i) eccentricity e = $\frac{1}{2}$ and foci (± 2, 0)
(ii) eccentricity e = $\frac{2}{3}$ and length of latus rectum = 5
(iii) eccentricity e = $\frac{1}{2}$ and semi-major axis = 4
(iv) eccentricity e = $\frac{1}{2}$ and major axis = 12
(v) The ellipse passes through (1, 4) and (−6, 1).
(vi) Vertices (± 5, 0), foci (± 4, 0)
(vii) Vertices (0, ± 13), foci (0, ± 5)
(viii) Vertices (± 6, 0), foci (± 4, 0)
(ix) Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)
(x) Ends of major axis (0, ± $\sqrt{5}$), ends of minor axis (± 1, 0)
(xi) Length of major axis 26, foci (± 5, 0)
(xii) Length of minor axis 16 foci (0, ± 6)
(xiii) Foci (± 3, 0), a = 4

#### Question 6:

Find the equation of the ellipse whose foci are (4, 0) and (−4, 0), eccentricity = 1/3.

#### Question 7:

Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus-rectum is 10.

#### Question 8:

Find the equation of the ellipse whose centre is (−2, 3) and whose semi-axis are 3 and 2 when major axis is (i) parallel to x-axis (ii) parallel to y-axis.

#### Question 9:

Find the eccentricity of an ellipse whose latus rectum is
(i) half of its minor axis
(ii) half of its major axis.

#### Question 10:

Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i) x2 + 2y2 − 2x + 12y + 10 = 0
(ii) x2 + 4y2 − 4x + 24y + 31 = 0
(iii) 4x2 + y2 − 8x + 2y + 1 = 0
(iv) 3x2 + 4y2 − 12x − 8y + 4 = 0
(v) 4x2 + 16y2 − 24x − 32y − 12 = 0
(vi) x2 + 4y2 − 2x = 0

#### Question 11:

Find the equation of an ellipse whose foci are at (± 3, 0) and which passes through (4, 1).

#### Question 12:

Find the equation of an ellipse whose eccentricity is 2/3, the latus-rectum is 5 and the centre is at the origin.

#### Question 13:

Find the equation of an ellipse with its foci on y-axis, eccentricity 3/4, centre at the origin and passing through (6, 4).

#### Question 14:

Find the equation of an ellipse whose axes lie along coordinate axes and which passes through (4, 3) and (−1, 4).

#### Question 15:

Find the equation of an ellipse whose axes lie along the coordinate axes, which passes through the point (−3, 1) and has eccentricity equal to $\sqrt{2/5}$.

#### Question 16:

Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directrices is 18 units.

#### Question 17:

Find the equation of an ellipse whose vertices are (0, ± 10) and eccentricity e = $\frac{4}{5}$.

#### Question 18:

A rod of length 12 m moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with x-axis.

Let AB be the rod making an angle θ with OX and let P (x, y) be the point on it such that AP = 3 cm.
Then, PB = AB – AP = (12 – 3) cm = 9 cm      [∵ AB = 12 cm]
From P, draw PQ⊥OY and PR⊥OX.

#### Question 19:

Find the equation of the set of all points whose distances from (0, 4) are $\frac{2}{3}$ of their distances from the line y = 9.

We have
$\mathrm{PQ}=\frac{2}{3}\mathrm{PL}\phantom{\rule{0ex}{0ex}}⇒\sqrt{{\left(x-0\right)}^{2}+{\left(y-4\right)}^{2}}=\frac{2}{3}\left(y-9\right)\phantom{\rule{0ex}{0ex}}⇒{3}^{2}\left[{x}^{2}+{\left(y-4\right)}^{2}\right]={2}^{2}{\left(y-9\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒9{x}^{2}+9{y}^{2}-72y+144=4{y}^{2}-72y+324\phantom{\rule{0ex}{0ex}}⇒9{x}^{2}+5{y}^{2}=180\phantom{\rule{0ex}{0ex}}⇒\frac{{x}^{2}}{20}+\frac{{y}^{2}}{36}=1$

#### Question 1:

If the lengths of semi-major and semi-minor axes of an ellipse are 2 and $\sqrt{3}$ and their corresponding equations are y − 5 = 0 and x + 3 = 0, then write the equation of the ellipse.

#### Question 2:

Write the eccentricity of the ellipse 9x2 + 5y2 − 18x − 2y − 16 = 0.

#### Question 3:

Write the centre and eccentricity of the ellipse 3x2 + 4y2 − 6x + 8y − 5 = 0.

#### Question 4:

PSQ is a focal chord of the ellipse 4x2 + 9y2 = 36 such that SP = 4. If S' is the another focus, write the value of S'Q.

#### Question 5:

Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.

#### Question 6:

If the distance between the foci of an ellipse is equal to the length of the latus-rectum, write the eccentricity of the ellipse.

#### Question 7:

If S and S' are two foci of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ and B is an end of the minor axis such that ∆BSS' is equilateral, then write the eccentricity of the ellipse.

#### Question 8:

If the minor axis of an ellipse subtends an equilateral triangle with vertex at one end of major axis, then write the eccentricity of the ellipse.

#### Question 9:

If a latus rectum of an ellipse subtends a right angle at the centre of the ellipse, then write the eccentricity of the ellipse.

#### Question 1:

For the ellipse 12x2 + 4y2 + 24x − 16y + 25 = 0

(a) centre is (−1, 2)

(b) lengths of the axes are $\sqrt{3}$ and 1

(c) eccentricity = $\sqrt{\frac{2}{3}}$

(d) all of these

#### Question 2:

The equation of the ellipse with focus (−1, 1), directrix xy + 3 = 0 and eccentricity 1/2 is
(a) 7x2 + 2xy + 7y2 + 10x + 10y + 7 = 0
(b) 7x2 + 2xy + 7y2 + 10x − 10y + 7 = 0
(c) 7x2 + 2xy + 7y2 + 10x − 10y − 7 = 0
(d) none of these

#### Question 3:

The equation of the circle drawn with the two foci of $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ as the end-points of a diameter is
(a) x2 + y2 = a2 + b2
(b) x2 + y2 = a2
(c) x2 + y2 = 2a2
(d) x2 + y2 = a2b2

#### Question 4:

The eccentricity of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ if its latus rectum is equal to one half of its minor axis, is
(a) $\frac{1}{\sqrt{2}}$

(b) $\frac{\sqrt{3}}{2}$

(c) $\frac{1}{2}$

(d) none of these

#### Question 5:

The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is
(a) $\frac{\sqrt{5}-1}{2}$

(b) $\frac{\sqrt{5}+1}{2}$

(c) $\frac{\sqrt{5}-1}{4}$

(d) none of these

#### Question 6:

The eccentricity of the ellipse, if the minor axis is equal to the distance between the foci, is
(a) $\frac{\sqrt{3}}{2}$

(b) $\frac{2}{\sqrt{3}}$

(c) $\frac{1}{\sqrt{2}}$

(d) $\frac{\sqrt{2}}{3}$

#### Question 7:

The difference between the lengths of the major axis and the latus-rectum of an ellipse is
(a) ae
(b) 2ae
(c) ae2
(d) 2ae2

#### Question 8:

The eccentricity of the conic 9x2 + 25y2 = 225 is
(a) 2/5
(b) 4/5
(c) 1/3
(d) 1/5
(e) 3/5

#### Question 9:

The latus-rectum of the conic 3x2 + 4y2 − 6x + 8y − 5 = 0 is

(a) 3

(b) $\frac{\sqrt{3}}{2}$

(c) $\frac{2}{\sqrt{3}}$

(d) none of these

#### Question 10:

The equations of the tangents to the ellipse 9x2 + 16y2 = 144 from the point (2, 3) are
(a) y = 3, x = 5
(b) x = 2, y = 3
(c) x = 3, y = 2
(d) x + y = 5, y = 3

#### Question 11:

The eccentricity of the ellipse 4x2 + 9y2 + 8x + 36y + 4 = 0 is

(a) $\frac{5}{6}$

(b) $\frac{3}{5}$

(c) $\frac{\sqrt{2}}{3}$

(d) $\frac{\sqrt{5}}{3}$

#### Question 12:

The eccentricity of the ellipse 4x2 + 9y2 = 36 is

(a) $\frac{1}{2\sqrt{3}}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{\sqrt{5}}{3}$

(d) $\frac{\sqrt{5}}{6}$

#### Question 13:

The eccentricity of the ellipse 5x2 + 9y2 = 1 is
(a) 2/3
(b) 3/4
(c) 4/5
(d) 1/2

#### Question 14:

For the ellipse x2 + 4y2 = 9
(a) the eccentricity is 1/2
(b) the latus-rectum is 3/2
(c) a focus is
(d) a directrix is x = $-2\sqrt{3}$

#### Question 15:

If the latus rectum of an ellipse is one half of its minor axis, then its eccentricity is

(a) $\frac{1}{2}$

(b) $\frac{1}{\sqrt{2}}$

(c) $\frac{\sqrt{3}}{2}$

(d) $\frac{\sqrt{3}}{4}$

#### Question 16:

An ellipse has its centre at (1, −1) and semi-major axis = 8 and it passes through the point
(1, 3). The equation of the ellipse is

(a) $\frac{{\left(x+1\right)}^{2}}{64}+\frac{{\left(y+1\right)}^{2}}{16}=1$

(b) $\frac{{\left(x-1\right)}^{2}}{64}+\frac{{\left(y+1\right)}^{2}}{16}=1$

(c) $\frac{{\left(x-1\right)}^{2}}{16}+\frac{{\left(y+1\right)}^{2}}{64}=1$

(d) $\frac{{\left(x+1\right)}^{2}}{64}+\frac{{\left(y-1\right)}^{2}}{16}=1$

#### Question 17:

The sum of the focal distances of any point on the ellipse 9x2 + 16y2 = 144 is
(a) 32
(b) 18
(c) 16
(d) 8

#### Question 18:

If (2, 4) and (10, 10) are the ends of a latus-rectum of an ellipse with eccentricity 1/2, then the length of semi-major axis is
(a) 20/3
(b) 15/3
(c) 40/3
(d) none of these

#### Question 19:

The equation $\frac{{x}^{2}}{2-\mathrm{\lambda }}+\frac{{y}^{2}}{\mathrm{\lambda }-5}+1=0$ represents an ellipse, if
(a) λ < 5
(b) λ < 2
(c) 2 < λ < 5
(d) λ < 2 or λ > 5

#### Question 20:

The eccentricity of the ellipse 9x2 + 25y2 − 18x − 100y − 116 = 0, is
(a) 25/16
(b) 4/5
(c) 16/25
(d) 5/4

#### Question 21:

If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to
(a) $\frac{1}{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{1}{\sqrt{2}}$

(d) $\frac{2\sqrt{2}}{3}$

(e) $\frac{2}{3\sqrt{2}}$

#### Question 22:

The eccentricity of the ellipse 25x2 + 16y2 = 400 is
(a) 3/5
(b) 1/3
(c) 2/5
(d) 1/5

#### Question 23:

The eccentricity of the ellipse 5x2 + 9y2 = 1 is
(a) 2/3
(b) 3/4
(c) 4/5
(d) 1/2

#### Question 24:

The eccentricity of the ellipse 4x2 + 9y2 = 36 is

(a) $\frac{1}{2\sqrt{3}}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{\sqrt{5}}{3}$

(d) $\frac{\sqrt{5}}{6}$