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#### 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:

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

#### Question 25:

The length of the latusrectum of the ellipse 3x2 + y2 = 12 is
(a) 4
(b) 3
(c) 8
(d) $\frac{4}{\sqrt{3}}$

Given equation of ellipse is

Hence, the correct answer is option D.

#### Question 26:

If e is the eccentricity of the ellipse then
(a) b2 = a2(1 – e2)
(b) a2 = b2(1 – e2)
(c) a2 = b2(e2 – 1)
(d) b2 = a2(e2 – 1)

For a < b,
Given equation of ellipse is $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$
Eccentricity $e=\sqrt{1-\frac{{a}^{2}}{{b}^{2}}}$

Hence, the correct answer is option B.

#### Question 27:

The equation of the ellipse whose centre is at the origin and the x-axis, the major axis, which asses through the points (–3, 1) and (2, –2) is
(a) 5x2 + 3y2 = 32
(b) 3x2 + 5y2 = 32
(c) 5x2 – 3y2 = 32
(d) 3x2 + 5y2 + 32 = 0

for an ellipse, centre is given as (0, 0) major axis as x axis
i.e equation is of the form
Since ellipses passes through (–3, 1) and (2, –2)
We get,

Hence, the correct answer is option C.

#### Question 1:

If the latusrectum of an ellipse be equal to half of its minor axis, then its eccentricity is ___________.

For ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ where a > b
Major axis is given by 2a and minor axis is 2b

Since latus rectum is
Given latus rectum is half of its minor axis
i.e $\frac{2{b}^{2}}{a}=\frac{1}{2}×2b$
i.e a = 2b

#### Question 2:

The eccentricity of the ellipse 16x2 + 7y2 = 112 is ___________.

For given ellipse 16x+ 7y2 =112

#### Question 3:

If the distance between the directrices of an ellipse be three times the distance between its foci, then the eccentricity is ___________.

for an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ where a > b
Distance between directrix is $\frac{2a}{e}$ where e is eccentricity and Distance between foci is 2ae
According to given condition,
distance between directrix = 3(distance between foci)

#### Question 4:

If the eccentricity of an ellipse is $\frac{5}{8}$ and the distance between its foci is 10, then the length of the latusrectum is ___________.

For an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$
eccentricity = $\frac{5}{8}$
Distance between foci is 10      (given)
i.e  2ae = 10
i.e  ae = 5
i.e  a$×\frac{5}{8}$ = 5
i.e  a = 8
Length of latus rectum is
Since   ${e}^{2}=1-{\left(\frac{b}{a}\right)}^{2}$
i.e ${b}^{2}={a}^{2}\left(1-{e}^{2}\right)$

#### Question 5:

If the coordinates of foci and vertices of an ellipse are (±1, 0) and (±2, 0) respectively, then the minor axis is of length ___________.

For an ellipse,
co-ordinates of foci is (±1, 0)
co-ordinates of vertices is (± 2, 0)
since y-co-ordinate is zero,
∴ equation of ellipse is of form

Since vertex in general is given by (± a, 0)
and foci is given by (± ae, 0)  ; where e is eccentricity.
a = 2 and ae = 1

#### Question 6:

The distance between the directrices of the ellipse $\frac{{x}^{2}}{36}+\frac{{y}^{2}}{20}=1$ is ___________.

Given parabola is $\frac{{x}^{2}}{36}+\frac{{y}^{2}}{20}=1$
i.e a2 = 36  and  b2 = 20

∴ equation of directrices is $x=\frac{a}{e}$ and $x=-\frac{a}{e}$
i.e
i.e x = 9 and x = $-$9 are equations of directrices
Distance between directrices is 18

#### Question 7:

The distance between the foci of the ellipse 3x2 + 4y2 = 48 is ___________.

Given ellipse is 3x2 + 4y2 = 48

∴    Distance of foci is 4

#### Question 8:

The eccentricity of the ellipse whose latusrectum is equal to the distance between the foci, is ___________.

For an ellipse, $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$
Given, latusrectum = Distance between foci
i.e $\frac{2{b}^{2}}{a}$  = 2ae
i.e 2b2 = 2a2e
i.e $\frac{{b}^{2}}{{a}^{2}}$= e

#### Question 9:

The equation $\frac{{x}^{2}}{2-r}+\frac{{y}^{2}}{r-5}+1=0$ represents an ellipse, if ___________.

For given equation  $\frac{{x}^{2}}{2-r}+\frac{{y}^{2}}{r-5}+1=0$
Represents an ellipse if
$-$ r < 0 and$-$ 50
i.e r > 2 and r < 5
i.e 2 < r < 5

#### Question 10:

An ellipse is described by using an endless string which passed over two points. If the axes are 6 cm and 4 cm, the length of the string and distance between the points are ___________.

Let equation of ellipse be $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$
Length of major axis i.e  2a = 6
i.e  a = 3
and Length of minor axis is 4 = 2b
i.e  b = 2
$\therefore$ eccentricity $e=1-\frac{{b}^{2}}{{a}^{2}}$
i.e $e=1-\frac{4}{9}=\frac{5}{9}$
Let S and S' represents the foci of ellipse and P be any point an ellipse
$\therefore$ SP + S'P = 2a
$\therefore$ Length of the endless string SP + S'P + SS'
= 2a + 2ae
$=2×3+2×3×\frac{\sqrt{5}}{3}$
i.e  Length of the endless string = 6 + $2\sqrt{5}$ and distance between the points is SS' = $2\sqrt{5}$

#### Question 11:

The distance between the directrices of the ellipse $\frac{{x}^{2}}{36}+\frac{{y}^{2}}{20}=1$ is ___________.

Given ellipse is of the form,
$\frac{{x}^{2}}{36}+\frac{{y}^{2}}{20}=1$
i.e a2 = 36  and  b2 = 20

∴ equation of directrices is $x=\frac{a}{e}$ and $x=-\frac{a}{e}$
i.e
i.e x = 9 and x = $-$9 are equations of directrices
Distance between directrices is 18

#### Question 12:

The equation of the ellipse having foci (0, 1),(0, –1) and minor axis of length 1 is ___________.

For an ellipse, length of minor axis is given = l and foci is given by = (0, $±$1)
i.e ellipse is of the form
i.e be $±$1
Also,  2a = 1   i.e $\frac{1}{a}$= 2
i.e  a$\frac{1}{2}$
and  e $±\frac{1}{b}$

Hence, equation of ellipse is 20x2 + 4y2 = 5.

#### Question 13:

The eccentricity of the ellipse is given by ___________.

For
eccentricity, $e=\sqrt{1-\frac{{a}^{2}}{{b}^{2}}}$

#### Question 14:

The length of the latusretum of the ellipse 3x2 + y2 = 12 is ___________.

Given ellipse is 3xy2 = 12

​i.e  a2 = 4 and b= 12
$\therefore$ Length of latus rectum is $\frac{2{b}^{2}}{a}$ =

#### Question 15:

If the latusrectum of an ellipse with axis along x-axis and centre at origin is 10, distance between foci = length of minor axis, then equation of the ellipse is ___________.

Centre of ellipse is given (0, 0)
Let the equation of ellipse be $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$
Latusrectum is given, which is 10

Also given, distance between foci = Length of minor axis
i.e 2ae = 2b
i.e ae = b
i.e e$\frac{b}{a}$
Since b2 = a2 (1 – e2)
i.e a2e2 = a(1 – e2)
i.e e2 = 1 – e2
i.e 2e2 = 1
i.e e$±\frac{1}{\sqrt{2}}$
i.e e$\frac{1}{\sqrt{2}}$
$⇒a=b\sqrt{2}$
Since   $\frac{2{b}^{2}}{a}=10$

$\therefore$ equation of ellipse is $\frac{{x}^{2}}{{\left(10\right)}^{2}}+\frac{{b}^{2}}{{\left(5\sqrt{2}\right)}^{2}}=1$
i.e $\frac{{x}^{2}}{100}+\frac{{b}^{2}}{50}=1$
i.e   x2 + 2b2 – 100
Hence equation of ellipse in x2 + 2b2 = 100

#### 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.