How many questions from Ellipse appear in JEE?

Ellipse has appeared in 103 of the last 186 JEE Main and JEE Advanced papers — about 55% of them — contributing 114 questions in total across those papers.

Is Ellipse an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 55% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Ellipse questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Ellipse JEE Previous Year Questions with Answers

Q1·MathematicsNumericalJEE Advanced 2026

Passage: Consider the ellipses given by

x^{2} + 4y^{2} = 1

and

4x^{2} + y^{2} = 1

. Question: Let

P

be the point in the first quadrant where the given ellipses intersect. If

\theta

is the acute angle between the tangents to the given ellipses at the point

P

, then the value of

4\tan\theta

is _____.

Q2·MathematicsSingle correctJEE Main 2026

Let

\dfrac{x^2}{f(a^2 + 7a + 3)} + \dfrac{y^2}{f(3a + 15)} = 1

represent an ellipse with major axis along

y

-axis, where

f

is a strictly decreasing positive function on

\mathbb{R}

. If the set of all possible values of

a

is

\mathbb{R} - [\alpha, \beta]

, then

\alpha^2 + \beta^2

is equal to:

(A) $28$
(B) $40$
(C) $61$
(D) $24$

Q3·MathematicsSingle correctJEE Main 2026

Let

x = 9

be a directrix of an ellipse

E

, whose centre is at the origin and eccentricity is

\frac{1}{3}

. Let

P(\alpha, 0)

,

\alpha > 0

, be a focus of

E

and

AB

be a chord passing through

P

. Then the locus of the mid point of

AB

is :

(A) $9y^{2} = 8x(1 - x)$
(B) $3y^{2} = 4x(1 - x)$
(C) $9y^{2} = 8x(x - 1)$
(D) $3y^{2} = 4x(x - 1)$

Q4·MathematicsSingle correctJEE Main 2026

Let a focus of the ellipse

E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

be

S(4,0)

and its eccentricity be

\frac{4}{5}

. If the point

P(3, \alpha)

lies on

E

and

O

is the origin, then the area of

\triangle POS

is equal to:

(A)12/5
(B)14/5
(C)24/5
(D)48/5

Q5·MathematicsNumericalJEE Main 2026

Consider the parabola

P : y^2 = 4kx

and the ellipse

E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

. Let the line segment joining the points of intersection of

P

and

E

, be their latus rectums. If the eccentricity of

E

is

e

, then

e^2 + 2\sqrt{2}

is equal to _______.

Q6·MathematicsSingle correctJEE Main 2026

Let an ellipse

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1

,

a<b

, pass through the point

(4,3)

and have eccentricity

\frac{\sqrt{5}}{3}

. Then the length of its latus rectum is :

(A) $\frac{4\sqrt{5}}{3}$
(B) $2\sqrt{5}$
(C) $\frac{7\sqrt{5}}{3}$
(D) $\frac{8\sqrt{5}}{3}$

Q7·MathematicsNumericalJEE Main 2026

Let A be the point

(3, 0)

and circles with variable diameter AB touch the circle

x^2 + y^2 = 36

internally. Let the curve C be the locus of the point B. If the eccentricity of C is

e

, then

72e^2

is equal to ________.

Q8·MathematicsSingle correctJEE Main 2026

An ellipse has its center at (1,-2), one focus at (3,-2) and one vertex at (5, - 2). Then the length of its latus rectum is :

(A) $\dfrac{16}{\sqrt{3}}$
(B)6
(C) $4\sqrt{3}$
(D) $6\sqrt{3}$

Q9·MathematicsSingle correctJEE Main 2026

Let each of the two ellipses

E_{1}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b)

and

E_{2}:\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1

,

(A<B)

have eccentricity

\frac{4}{5}

. Let the lengths of the latus recta of

E_{1}

and

E_{2}

be

\ell_{1}

and

\ell_{2}

, respectively, such that

2\ell_{1}^{2}=9\ell_{2}

. If the distance between the foci of

E_{1}

is 8, then the distance between the foci of

E_{2}

is

(A) $\frac{96}{5}$
(B) $\frac{32}{5}$
(C) $\frac{16}{5}$
(D) $\frac{8}{5}$

Q10·MathematicsNumericalJEE Main 2026

Let (h, k) lie on the circle

C : x^{2} + y^{2} = 4

and the point

(2h + 1, 3k + 2)

lie on an ellipse with eccentricity e. Then the value of

\dfrac{5}{e^{2}}

is equal to ________.

Q11·MathematicsSingle correctJEE Main 2026

Let the length of the latus rectum of an ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

,

(a > b)

, be 30. If its eccentricity is the maximum value of the function

f(t) = -\frac{3}{4} + 2t - t^{2}

, then

(a^{2} + b^{2})

is equal to -

(A)516
(B)256
(C)496
(D)276

Q12·MathematicsSingle correctJEE Main 2026

Let the line y - x = 1 intersect the ellipse

\frac{x^2}{2} + \frac{y^2}{1} = 1

at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

(A) $\pi - \tan^{-1}\left(\frac{1}{4}\right)$
(B) $\frac{\pi}{2} + \tan^{-1}\left(\frac{1}{4}\right)$
(C) $\frac{\pi}{2} + 2\tan^{-1}\left(\frac{1}{4}\right)$
(D) $\frac{\pi}{2} - \tan^{-1}\left(\frac{1}{4}\right)$

Q13·MathematicsSingle correctJEE Main 2026

Let S and S' be the foci of the ellipse

\frac{x^2}{25} + \frac{y^2}{9} = 1

and P (α, β) be a point on the ellipse in the first quadrant. If

(SP)^2 + (S'P)^2 - SP \bullet S'P = 37

, then

α^2 + β^2

is equal to :

(A)15
(B)11
(C)17
(D)13

Q14·MathematicsSingle correctJEE Main 2026

In the line

\alpha x + 4y = \sqrt{7}

, where α ∈ R, touches the ellipse

3x^2 + 4y^2 = 1

at the point P in the first quadrant, then one of the focal distances of P is :

(A) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{11}}$
(B) $\frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{5}}$
(C) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{5}}$
(D) $\frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{7}}$

Q15·MathematicsMultiple correctJEE Advanced 2025

Let

P(x_1, y_1)

and

Q(x_2, y_2)

be two distinct points on the ellipse

\frac{x^2}{9} + \frac{y^2}{4} = 1

such that

y_1 > 0

and

y_2 > 0

. Let C denote the circle

x^2 + y^2 = 9

, and M be the point (3, 0). Suppose the line

x = x_1

intersects C at R, and the line

x = x_2

intersects C at S, such that the y-coordinates of R and S are positive. Let

\angle ROM = \frac{\pi}{6}

and

\angle SOM = \frac{\pi}{3}

, where O denotes the origin (0, 0). Let |XY| denote the length of the line segment XY. Then which of the following statements is (are) TRUE?

(A)The equation of the line joining P and Q is $2x + 3y = 3\left(1 + \sqrt{3}\right)$
(B)The equation of the line joining P and Q is $2x + y = 3\left(1 + \sqrt{3}\right)$
(C)If $N_2 = (x_2, 0)$ , then $3|N_2Q| = 2|N_2S|$
(D)If $N_1 = (x_1, 0)$ , then $9|N_1P| = 4|N_1R|$

Q16·MathematicsSingle correctJEE Main 2025

Let

e_1

and

e_2

be the eccentricities of the ellipse

\dfrac{x^2}{b^2}+\dfrac{y^2}{25}=1

and the hyperbola

\dfrac{x^2}{16}-\dfrac{y^2}{b^2}=1

, respectively. If

b<5

and

e_1e_2=1

, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

(A) $\dfrac{4}{5}$
(B) $\dfrac{3}{5}$
(C) $\dfrac{\sqrt{7}}{4}$
(D) $\dfrac{\sqrt{3}}{2}$

Q17·MathematicsSingle correctJEE Main 2025

Let the length of a latus rectum of an ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

be 10. If its eccentricity is the minimum value of the function

f(t)=t^2+t+\dfrac{11}{12}

,

t\in\mathbb{R}

, then

a^2+b^2

is equal to:

(A)125
(B)126
(C)120
(D)115

Q18·MathematicsSingle correctJEE Main 2025

The length of the latus-rectum of the ellipse, whose foci are

(2,5)

and

(2,-3)

and eccentricity is

\dfrac{4}{5}

, is

(A) $\dfrac{6}{5}$
(B) $\dfrac{50}{3}$
(C) $\dfrac{10}{3}$
(D) $\dfrac{18}{5}$

Q19·MathematicsSingle correctJEE Main 2025

Let for two distinct values of

p

the lines

y=x+p

touch the ellipse

E:\dfrac{x^2}{4^2}+\dfrac{y^2}{3^2}=1

at the points

A

and

B

. Let the line

y=x

intersect

E

at the points

C

and

D

. Then the area of the quadrilateral

ABCD

is equal to:

(A)36
(B)24
(C)48
(D)20

Q20·MathematicsSingle correctJEE Main 2025

The centre of a circle

C

is at the centre of the ellipse

E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

a>b

. Let

C

pass through the foci

F_1

and

F_2

of

E

such that the circle

C

and the ellipse

E

intersect at four points. Let

P

be one of these four points. If the area of the triangle

PF_1F_2

is 30 and the length of the major axis of

E

is 17, then the distance between the foci of

E

is:

(A)26
(B)13
(C)12
(D) $\dfrac{13}{2}$

Q21·MathematicsSingle correctJEE Main 2025

Let

C

be the circle of minimum area enclosing the ellipse

E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

with eccentricity

\dfrac{1}{2}

and foci

(\pm 2,0)

. Let

PQR

be a variable triangle, whose vertex

P

is on the circle

C

and the side

QR

of length

2a

is parallel to the major axis of

E

and contains the point of intersection of

E

with the negative

y

-axis. Then the maximum area of the triangle

PQR

is:

(A) $6(3+\sqrt2)$
(B) $8(3+\sqrt2)$
(C) $6(2+\sqrt3)$
(D) $8(2+\sqrt3)$

Q22·MathematicsSingle correctJEE Main 2025

A line passing through the point

P(\sqrt5,\sqrt5)

intersects the ellipse

\dfrac{x^2}{36}+\dfrac{y^2}{25}=1

at

A

and

B

such that

(PA)\cdot(PB)

is maximum. Then

5(PA^2+PB^2)

is equal to:

(A)218
(B)377
(C)290
(D)338

Q23·MathematicsSingle correctJEE Main 2025

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

(A) $\dfrac{4}{\sqrt{17}}$
(B) $\dfrac{\sqrt3}{16}$
(C) $\dfrac{3}{\sqrt{19}}$
(D) $\dfrac{\sqrt5}{7}$

Q24·MathematicsSingle correctJEE Main 2025

If

S

and

S'

are the foci of the ellipse

\dfrac{x^2}{18}+\dfrac{y^2}{9}=1

and

P

be a point on the ellipse, then

\min(SP\cdot S'P)+\max(SP\cdot S'P)

is equal to:

(A) $3(1+\sqrt{2})$
(B) $3(6+\sqrt{2})$
(C)9
(D)27

Q25·MathematicsSingle correctJEE Main 2025

If

\alpha x+\beta y=109

is the equation of the chord of the ellipse

\dfrac{x^2}{9}+\dfrac{y^2}{4}=1

, whose mid point is

\left(\dfrac{5}{2},\dfrac{1}{2}\right)

, then

\alpha+\beta

is equal to

(A)37
(B)46
(C)58
(D)72

Q26·MathematicsSingle correctJEE Main 2025

Let the ellipse

E_1:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

a>b

and

E_2:\dfrac{x^2}{A^2}+\dfrac{y^2}{B^2}=1

,

A<B

have same eccentricity

\dfrac{1}{\sqrt{3}}

. Let the product of their lengths of latus rectums be

\dfrac{32}{\sqrt{3}}

, and the distance between the foci of

E_1

be 4. If

E_1

and

E_2

meet at A, B, C and D, then the area of the quadrilateral ABCD equals:

(A) $6\sqrt{6}$
(B) $\dfrac{18\sqrt{6}}{5}$
(C) $\dfrac{12\sqrt{6}}{5}$
(D) $\dfrac{24\sqrt{6}}{5}$

Q27·MathematicsSingle correctJEE Main 2025

If the midpoint of a chord of the ellipse

\frac{x^2}{9}+\frac{y^2}{4}=1

is

\left(\sqrt{2},\frac{4}{3}\right)

, and the length of the chord is

\frac{2\sqrt{\alpha}}{3}

, then

\alpha

is:

(A)18
(B)22
(C)26
(D)20

Q28·MathematicsIntegerJEE Main 2025

Let

E_1:\frac{x^2}{9}+\frac{y^2}{4}=1

be an ellipse. Ellipses

E_i

's are constructed such that their centres and eccentricities are same as that of

E_1

, and the length of minor axis of

E_i

is the length of major axis of

E_{i+1}

(i\ge1)

. If

A_i

is the area of the ellipse

E_i

, then

\frac{5}{\pi}\left(\sum_{i=1}^{\infty} A_i\right)

, is equal to ____.

Q29·MathematicsSingle correctJEE Main 2025

Let the product of the focal distances of the point

\left(\sqrt3,\dfrac{1}{2}\right)

on the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

(a>b)

, be

\dfrac{7}{4}

. Then the absolute difference of the eccentricities of two such ellipses is

(A) $\dfrac{3-2\sqrt2}{3\sqrt2}$
(B) $\dfrac{1-\sqrt3}{\sqrt2}$
(C) $\dfrac{3-2\sqrt2}{2\sqrt3}$
(D) $\dfrac{1-\sqrt2}{\sqrt3}$

Q30·MathematicsSingle correctJEE Main 2025

The equation of the chord, of the ellipse

\dfrac{x^2}{25}+\dfrac{y^2}{16}=1

, whose mid-point is

(3,1)

is :

(A) $48x+25y=169$
(B) $4x+122y=134$
(C) $25x+101y=176$
(D) $5x+16y=31$

Q31·MathematicsSingle correctJEE Main 2025

The length of the chord of the ellipse

\dfrac{x^2}{4}+\dfrac{y^2}{2}=1

, whose mid-point is

\left(1,\dfrac{1}{2}\right)

, is :

(A) $\dfrac{2}{3}\sqrt{15}$
(B) $\dfrac{5}{3}\sqrt{15}$
(C) $\dfrac{1}{3}\sqrt{15}$
(D) $\sqrt{15}$

Q32·MathematicsSingle correctJEE Advanced 2024

Consider the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

. Let

S(p, q)

be a point in the first quadrant such that

\frac{p^{2}}{9} + \frac{q^{2}}{4} > 1

. Two tangents are drawn from

S

to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point

T

in the fourth quadrant. Let

R

be the vertex of the ellipse with positive x-coordinate and

O

be the centre of the ellipse. If the area of the triangle

\triangle ORT

is

\frac{3}{2}

, then which of the following options is correct ?

(A) $q = 2, p = 3\sqrt{3}$
(B) $q = 2, p = 4\sqrt{3}$
(C) $q = 1, p = 5\sqrt{3}$
(D) $q = 1, p = 6\sqrt{3}$

Q33·MathematicsSingle correctJEE Main 2024

Let the line

2x + 3y - k = 0

,

k > 0

, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is

x^2 + y^2 - 3x - 2y = 0

and the length of the latus rectum of the ellipse

x^2 + 9y^2 = k^2

is

\dfrac{m}{n}

, where m and n are coprime, then

2m + n

is equal to:

(A) $10$
(B) $11$
(C) $13$
(D) $12$

Q34·MathematicsSingle correctJEE Main 2024

Let P be a point on the ellipse

\dfrac{x^2}{9}+\dfrac{y^2}{4}=1

. Let the line passing through P and parallel to y-axis meet the circle

x^2+y^2=9

at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that

PR:RQ=4:3

as P moves on the ellipse, is:

(A) $\dfrac{11}{19}$
(B) $\dfrac{13}{21}$
(C) $\dfrac{\sqrt{139}}{23}$
(D) $\dfrac{\sqrt{13}}{7}$

Q35·MathematicsSingle correctJEE Main 2024

Let

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

a>b

, be an ellipse, whose eccentricity is

\dfrac{1}{\sqrt2}

and the length of the latus rectum is

\sqrt{14}

. Then the square of the eccentricity of

\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1

is:

(A) $3$
(B) $\dfrac72$
(C) $\dfrac32$
(D) $\dfrac52$

Q36·MathematicsSingle correctJEE Main 2024

For

0<\theta<\dfrac\pi2

, if the eccentricity of the hyperbola

x^2-y^2\csc^2\theta=5

is

\sqrt7

times the eccentricity of the ellipse

x^2\csc^2\theta+y^2=5

, then the value of

\theta

is:

(A) $\dfrac\pi6$
(B) $\dfrac{5\pi}{12}$
(C) $\dfrac\pi3$
(D) $\dfrac\pi4$

Q37·MathematicsNumericalJEE Main 2024

Let the foci and length of the latus rectum of an ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

a>b

be

(\pm 5,0)

and

\sqrt{50}

respectively. Then, the square of the eccentricity of the hyperbola

\dfrac{x^2}{b^2}-\dfrac{y^2}{a^2 b^2}=1

equals ______.

Q38·MathematicsSingle correctJEE Main 2024

If the foci of a hyperbola are same as that of the ellipse

\dfrac{x^2}{9}+\dfrac{y^2}{25}=1

and the eccentricity of the hyperbola is

\dfrac{15}{8}

times the eccentricity of the ellipse, then the smaller focal distance of the point

\left(\sqrt{2},\dfrac{14}{3}\sqrt{\dfrac{2}{5}}\right)

on the hyperbola, is equal to

(A) $7\sqrt{\dfrac{2}{5}}-\dfrac{8}{3}$
(B) $14\sqrt{\dfrac{2}{5}}-\dfrac{4}{3}$
(C) $14\sqrt{\dfrac{2}{5}}-\dfrac{16}{3}$
(D) $7\sqrt{\dfrac{2}{5}}+\dfrac{8}{3}$

Q39·MathematicsSingle correctJEE Main 2024

Let

A(\alpha,0)

and

B(0,\beta)

be the points on the line

5x+7y=50

. Let the point

P

divide the line segment

AB

internally in the ratio

7:3

. Let

3x-25=0

be a directrix of the ellipse

E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

and the corresponding focus be

S

. If from

S

, the perpendicular on the x-axis passes through

P

, then the length of the latus rectum of

E

is equal to:

(A) $\dfrac{25}{3}$
(B) $\dfrac{32}{9}$
(C) $\dfrac{25}{9}$
(D) $\dfrac{32}{5}$

Q40·MathematicsSingle correctJEE Main 2024

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

(A) $\dfrac{\sqrt5}{3}$
(B) $\dfrac{\sqrt3}{2}$
(C) $\dfrac{1}{\sqrt3}$
(D) $\dfrac{2}{\sqrt5}$

Q41·MathematicsNumericalJEE Main 2024

If the points of intersection of two distinct conics

x^2+y^2=4b

and

\dfrac{x^2}{16}+\dfrac{y^2}{b^2}=1

lie on the curve

y^2=3x^2

, then

3\sqrt{3}

times the area of the rectangle formed by the intersection points is ______.

Q42·MathematicsSingle correctJEE Main 2024

Let

e_1

be the eccentricity of the hyperbola

\dfrac{x^2}{16}-\dfrac{y^2}{9}=1

and

e_2

be the eccentricity of the ellipse

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

,

a>b

, which passes through the foci of the hyperbola. If

e_1 e_2=1

, then the length of the chord of the ellipse parallel to the x-axis and passing through

(0,2)

is :

(A) $4\sqrt5$
(B) $\dfrac{8\sqrt5}{3}$
(C) $\dfrac{10\sqrt5}{3}$
(D) $3\sqrt5$

Q43·MathematicsSingle correctJEE Main 2024

The length of the chord of the ellipse

\dfrac{x^2}{25}+\dfrac{y^2}{16}=1

, whose mid point is

\left(1,\dfrac{2}{5}\right)

, is equal to:

(A) $\dfrac{\sqrt{1691}}{5}$
(B) $\dfrac{\sqrt{2009}}{5}$
(C) $\dfrac{\sqrt{1741}}{5}$
(D) $\dfrac{\sqrt{1541}}{5}$

Q44·MathematicsMultiple correctJEE Advanced 2023

Let

T_1

and

T_2

be two distinct common tangents to the ellipse

E : \frac{x^2}{6} + \frac{y^2}{3} = 1

and the parabola

P : y^2 = 12x

. Suppose that the tangent

T_1

touches

P

and

E

at the points

A_1

and

A_2

, respectively and the tangent

T_2

touches

P

and

E

at the points

A_4

and

A_3

, respectively. Then which of the following statements is(are) true?

(A)The area of the quadrilateral $A_1A_2A_3A_4$ is 35 square units
(B)The area of the quadrilateral $A_1A_2A_3A_4$ is 36 square units
(C)The tangents $T_1$ and $T_2$ meet the x -axis at the point $(-3, 0)$
(D)The tangents $T_1$ and $T_2$ meet the x -axis at the point $(-6, 0)$

Q45·MathematicsNumericalJEE Main 2023

Let an ellipse with centre

(1, 0)

and latus rectum of length

\frac{1}{2}

have its major axis along

x

-axis. If its minor axis subtends an angle

60^\circ

at the foci, then the square of the sum of the lengths of its minor and major axes is equal to____

Q46·MathematicsSingle correctJEE Main 2023

Let the tangent and normal at the point

(3\sqrt3,1)

on the ellipse

\dfrac{x^2}{36}+\dfrac{y^2}{4}=1

meet the

y

-axis at the points

A

and

B

respectively. Let the circle

C

be drawn taking

AB

as a diameter and the line

x=2\sqrt5

intersect

C

at the points

P

and

Q

. If the tangents at the points

P

and

Q

on the circle intersect at the point

(\alpha,0)

and

(\beta,0)

, then

\alpha^2-\beta^2

is equal to:

(A) $\dfrac{314}{5}$
(B) $\dfrac{304}{5}$
(C) $60$
(D) $61$

Q47·MathematicsSingle correctJEE Main 2023

Let P

\left(\dfrac{2\sqrt3}{\sqrt7},\dfrac{6}{\sqrt7}\right)

, Q, R and S be four points on the ellipse

9x^2+4y^2=36

. Let PQ and RS be mutually perpendicular and pass through the origin. If

\dfrac{1}{(PQ)^2}+\dfrac{1}{(RS)^2}=\dfrac{p}{q}

, where p and q are coprime, then

p+q

is equal to

(A)143
(B)157
(C)137
(D)147

Q48·MathematicsSingle correctJEE Main 2023

If the radius of the largest circle with centre

(2,0)

inscribed in the ellipse

x^2+4y^2=36

is

r

, then

12r^2

is equal to

(A) $72$
(B) $115$
(C) $92$
(D) $69$

Q49·MathematicsSingle correctJEE Main 2023

Consider ellipses

E_k : kx^2 + k^2 y^2 = 1

,

k = 1, 2, \ldots, 20

. Let

C_k

be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse

E_k

. If

r_k

is the radius of the circle

C_k

, then the value of

\sum_{k=1}^{20} \frac{1}{r_k^2}

is

(A)3080
(B)3210
(C)3320
(D)2870

Q50·MathematicsSingle correctJEE Main 2023

Let a circle of radius

4

be concentric to the ellipse

15x^2+19y^2=285

. Then the common tangents are inclined to the minor axis of the ellipse at the angle:

(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{12}$
(D) $\frac{\pi}{6}$

Q51·MathematicsSingle correctJEE Main 2023

Let the ellipse E:

x^2+9y^2=9

intersect the positive x- and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is

\dfrac{m}{n}

, where m and n are coprime, then

m-n

is equal to

(A) $18$
(B) $16$
(C) $17$
(D) $15$

Q52·MathematicsNumericalJEE Main 2023

Let the eccentricity of an ellipse

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

be reciprocal to that of the hyperbola

2x^2-2y^2=1

. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is

Q53·MathematicsSingle correctJEE Main 2023

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is

a

and the number of persons who speak only Hindi is

b

, then the eccentricity of the ellipse

25\left(\frac{x^2}{b^2}+\frac{y^2}{a^2}\right)=1

is equal to

(A) $\frac{3\sqrt{15}}{12}$
(B) $\frac{\sqrt{117}}{12}$
(C) $\frac{\sqrt{119}}{12}$
(D) $\frac{\sqrt{129}}{12}$

Q54·MathematicsNumericalJEE Main 2023

The line

x=8

is the directrix of the ellipse

E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1

with the corresponding focus

(2,0)

. If the tangent to E at the point P in the first quadrant passes through the point

(0,4\sqrt3)

and intersects the x-axis at Q, then

(3PQ)^2

equal to

Q55·MathematicsSingle correctJEE Main 2023

If the maximum distance of normal to the ellipse

\frac{x^2}{4}+\frac{y^2}{b^2}=1,\ b<2

, from the origin is

1

, then the eccentricity of the ellipse is :

(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{4}$
(C) $\frac{\sqrt{3}}{2}$
(D) $\frac{1}{\sqrt{2}}$

Q56·MathematicsNumericalJEE Main 2023

Let a tangent to the curve

9x^2+16y^2=144

intersect the coordinate axes at the points

A

and

B

. Then, the minimum length of the line segment

AB

is _______ .

Q57·MathematicsNumericalJEE Main 2023

Let

C

be the largest circle centred at

(2,0)

and inscribed in the ellipse

\dfrac{x^2}{36}+\dfrac{y^2}{16}=1

. If

(1,\alpha)

lies on

C

, then

10\alpha^2

is equal to _______ .

Q58·MathematicsSingle correctJEE Advanced 2022

Consider the ellipse

\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1

. Let

H\left(\alpha, 0\right)

,

0 < \alpha < 2

, be a point. A straight line drawn through

H

parallel to

y

-axis crosses the ellipse and its auxiliary circle at points

E

and

F

respectively, in the first quadrant. The tangents to the ellipse at the point

E

intersects the positive

x

-axis at a point

G

. Suppose the straight line joining

F

and the origin makes an angle

\phi

with the positive

x

-axis. The correct option is:

List-I	List-II
I.If $\phi = \frac{\pi}{4}$ , then the area of the triangle $FGH$ is	P. $\frac{\left(\sqrt{3} - 1\right)^{4}}{8}$
II.If $\phi = \frac{\pi}{3}$ , then the area of the triangle $FGH$ is	Q. $1$
III.If $\phi = \frac{\pi}{6}$ , then the area of the triangle $FGH$ is	R. $\frac{3}{4}$
IV.If $\phi = \frac{\pi}{12}$ , then the area of the triangle $FGH$ is	S. $\frac{1}{2\sqrt{3}}$
	T. $\frac{3\sqrt{3}}{2}$

(A)(I) → (R); (II) → (S); (III) → (Q); (IV) → (P)
(B)(I) → (R); (II) → (T); (III) → (S); (IV) → (P)
(C)(I) → (Q); (II) → (T); (III) → (S); (IV) → (P)
(D)(I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)

Q59·MathematicsNumericalJEE Main 2022

Let

S = \left\{ (x, y) \in \mathbb{N} \times \mathbb{N} : 9(x - 3)^2 + 16(y - 4)^2 \le 144 \right\}

and

T = \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : (x - 7)^2 + (y - 4)^2 \le 36 \right\}

. The

n(S \cap T)

is equal to______.

Q60·MathematicsSingle correctJEE Main 2022

Let a line L pass through the point of intersection of the lines

bx + 10y - 8 = 0

and

2x - 3y = 0

,

b \in R - \left\{\frac{4}{3}\right\}

. If the line L also passes through the point

(1, 1)

and touches the circle

17\,(x^{2} + y^{2}) = 16

, then the eccentricity of the ellipse

\frac{x^{2}}{5} + \frac{y^{2}}{b^{2}} = 1

is :

(A) $\frac{2}{\sqrt{5}}$
(B) $\sqrt{\frac{3}{5}}$
(C) $\frac{1}{\sqrt{5}}$
(D) $\sqrt{\frac{2}{5}}$

Q61·MathematicsNumericalJEE Main 2022

Let the tangents at the points P and Q on the ellipse

\frac{x^{2}}{2}+\frac{y^{2}}{4}=1

meet at the point

R\left(\sqrt{2},\,2\sqrt{2}-2\right)

. If S is the focus of the ellipse on its negative major axis, then

SP^{2} + SQ^{2}

is equal to

Q62·MathematicsNumericalJEE Main 2022

For the hyperbola H :

x^{2} - y^{2} = 1

and the ellipse E :

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

, a > b > 0, let the (1) eccentricity of E be reciprocal of the eccentricity of H, and (2) the line

y = \sqrt{\frac{5}{2}}x + K

be a common tangent of E and H. Then

4(a^{2} + b^{2})

is equal to ________.

Q63·MathematicsNumericalJEE Main 2022

An ellipse

E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1

passes through the vertices of the hyperbola

H:\frac{x^{2}}{49}-\frac{y^{2}}{64}=-1

. Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H. Let the product of the eccentricities of E and H be

\frac{1}{2}

. If

l

is the length of the latus rectum of the ellipse E, then the value of

113l

is equal to ______.

Q64·MathematicsNumericalJEE Main 2022

If the length of the latus rectum of the ellipse

x^{2}+4y^{2}+2x+8y-\lambda=0

is 4, and

l

is the length of its major axis, then

\lambda+l

is equal to ______.

Q65·MathematicsSingle correctJEE Main 2022

The acute angle between the pair of tangents drawn to the ellipse

2x^{2}+3y^{2}=5

from the point

(1,3)

is

(A) $\tan^{-1}\left(\frac{16}{7\sqrt{5}}\right)$
(B) $\tan^{-1}\left(\frac{24}{7\sqrt{5}}\right)$
(C) $\tan^{-1}\left(\frac{32}{7\sqrt{5}}\right)$
(D) $\tan^{-1}\left(\frac{3+8\sqrt{5}}{35}\right)$

Q66·MathematicsSingle correctJEE Main 2022

Let PQ be a focal chord of the parabola

y^2 = 4x

such that it subtends an angle of

\frac{\pi}{2}

at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse E:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a^2 > b^2

. If e is the eccentricity of the ellipse E, then the value of

\frac{1}{e^2}

is equal to :

(A) $1 + \sqrt{2}$
(B) $3 + 2\sqrt{2}$
(C) $1 + 2\sqrt{3}$
(D) $4 + 5\sqrt{3}$

Q67·MathematicsSingle correctJEE Main 2022

Let the eccentricity of an ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b,

be

\frac{1}{4}

. If this ellipse passes through the point

\left(-4\sqrt{\frac{2}{5}}, 3\right)

, then

a^{2} + b^{2}

is equal to :

(A)29
(B)31
(C)32
(D)34

Q68·MathematicsSingle correctJEE Main 2022

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse

x^2+2y^2=4

is an ellipse with eccentricity :

(A) $\frac{\sqrt{3}}{2}$
(B) $\frac{1}{2\sqrt{2}}$
(C) $\frac{1}{\sqrt{2}}$
(D) $\frac{1}{2}$

Q69·MathematicsNumericalJEE Main 2022

Let the common tangents to the curves

4(x^{2} + y^{2}) = 9

and

y^{2} = 4x

intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If

e

and

l

respectively denote the eccentricity and the length of the latus rectum of this ellipse, then

\frac{l}{e^{2}}

is equal to____________.

Q70·MathematicsSingle correctJEE Main 2022

If m is the slope of a common tangent to the curves

\frac{x^2}{16}+\frac{y^2}{9}=1

and

x^2+y^2=12

, then

12m^2

is equal to :

(A)6
(B)9
(C)10
(D)12

Q71·MathematicsSingle correctJEE Main 2022

The line

y = x + 1

meets the ellipse

\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1

at two points P and Q. If r is the radius of the circle with PQ as diameter then

(3r)^{2}

is equal to

(A)20
(B)12
(C)11
(D)8

Q72·MathematicsSingle correctJEE Main 2022

Let the maximum area of the triangle that can be inscribed in the ellipse

\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{4}=1

,

a > 2

, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be

6\sqrt{3}

. Then the eccentricity of the ellispe is :

(A) $\dfrac{\sqrt{3}}{2}$
(B) $\dfrac{1}{2}$
(C) $\dfrac{1}{\sqrt{2}}$
(D) $\dfrac{\sqrt{3}}{4}$

Q73·MathematicsIntegerJEE Advanced 2021

Let E be the ellipse

\frac{x^2}{16} + \frac{y^2}{9} = 1

. For any three distinct points P, Q and Q' on E, let M (P, Q) be the mid-point of the line segment joining P and Q, and M (P, Q') be the mid-point of the line segment joining P and Q'. Then the maximum possible value of the distance between M(P, Q) and M(P, Q'), as P, Q and Q' vary on E, is _____.

Q74·MathematicsSingle correctJEE Main 2021

Let θ be the acute angle between the tangents to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{1} = 1

and the circle

x^{2} + y^{2} = 3

at their point of intersection in the first quadrant. Then tanθ is equal to :

(A) $\frac{5}{2\sqrt{3}}$
(B) $\frac{2}{\sqrt{3}}$
(C) $\frac{4}{\sqrt{3}}$
(D)2

Q75·MathematicsSingle correctJEE Main 2021

The locus of mid-points of the line segments joining

(-3, -5)

and the points on the ellipse

\frac{x^2}{4} + \frac{y^2}{9} = 1

is :

(A) $9x^2 + 4y^2 + 18x + 8y + 145 = 0$
(B) $36x^2 + 16y^2 + 90x + 56y + 145 = 0$
(C) $36x^2 + 16y^2 + 108x + 80y + 145 = 0$
(D) $36x^2 + 16y^2 + 72x + 32y + 145 = 0$

Q76·MathematicsSingle correctJEE Main 2021

The line

12x \cos\theta + 5y \sin\theta = 60

is tangent to which of the following curves?

(A) $x^{2} + y^{2} = 169$
(B) $144x^{2} + 25y^{2} = 3600$
(C) $25x^{2} + 12y^{2} = 3600$
(D) $x^{2} + y^{2} = 60$

Q77·MathematicsSingle correctJEE Main 2021

On the ellipse

\frac{x^2}{8} + \frac{y^2}{4} = 1

let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line

x + 2y = 0

. Let S and S' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of

(5 - e^2)

. A is :

(A)6
(B)12
(C)14
(D)24

Q78·MathematicsNumericalJEE Main 2021

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3,−4), one focus at (4,−4) and one vertex at (5,−4) . If mx − y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m

^2

is equal to……………

Q79·MathematicsSingle correctJEE Main 2021

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of any ellipse with eccentricity

\frac{1}{3}

and the distance of the nearer focus from this directrix is

\frac{8}{\sqrt{53}}

, then the equation of the other directrix can be :

(A) $2x - 7y - 39 = 0$ or $\ 2x - 7y - 7 = 0$
(B) $11x + 7y + 8 = 0$ or $\ 11x + 7y - 15 = 0$
(C) $2x - 7y + 29 = 0$ or $\ 2x - 7y - 7 = 0$
(D) $11x - 7y - 8 = 0$ or $\ 11x + 7y + 15 = 0$

Q80·MathematicsSingle correctJEE Main 2021

If a tangent to the ellipse

x^{2}+4y^{2}=4

meets the tangents at the extremities of its major axis at B and C, then the circle with BC as diameter passes through the point :

(A) $(1,1)$
(B) $(\sqrt{2},0)$
(C) $(\sqrt{3},0)$
(D) $(-1,1)$

Q81·MathematicsSingle correctJEE Main 2021

Let an ellipse

E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}

, passes through

\left(\sqrt{\frac{3}{2}},1\right)

and has eccentricity

\frac{1}{\sqrt{3}}

. If a circle, centered at focus

F(\alpha,0), \alpha>0

, of E and radius

\frac{2}{\sqrt{3}}

, intersects E at two points P and Q, then

PQ^{2}

is equal to :

(A) $\frac{16}{3}$
(B)3
(C) $\frac{4}{3}$
(D) $\frac{8}{3}$

Q82·MathematicsSingle correctJEE Main 2021

Let

E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

,

a > b

. Let

E_2

be another ellipse such that it touches the end points of major axis of

E_1

and the foci of

E_2

are the end points of minor axis of

E_1

. If

E_1

and

E_2

have same eccentricities, then its value is :

(A) $\dfrac{-1 + \sqrt{5}}{2}$
(B) $\dfrac{-1 + \sqrt{3}}{2}$
(C) $\dfrac{-1 + \sqrt{8}}{2}$
(D) $\dfrac{-1 + \sqrt{6}}{2}$

Q83·MathematicsSingle correctJEE Main 2021

Let a tangent be drawn to the ellipse

\frac{x^{2}}{27} + y^{2} = 1

at

\left(3\sqrt{3}\cos\theta, \sin\theta\right)

where

\theta \in \left(0, \frac{\pi}{2}\right)

. Then the value of θ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

(A) $\frac{\pi}{8}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{3}$

Q84·MathematicsSingle correctJEE Main 2021

Let L be a tangent line to the parabola

y^{2} = 4x - 20

at

(6, 2)

. If L is also a tangent to the ellipse

\frac{x^{2}}{2}+\frac{y^{2}}{b} = 1

, then the value of

b

is equal to :

(A) $11$
(B) $14$
(C) $16$
(D) $20$

Q85·MathematicsSingle correctJEE Main 2021

If the point of intersections of the ellipse

\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1

and the circle

x^{2}+y^{2}=4b

,

b > 4

lie on the curve

y^{2}=3x^{2}

, then b is equal to:

(A) $12$
(B) $5$
(C) $6$
(D) $10$

Q86·MathematicsNumericalJEE Main 2021

Let L be a common tangent line to the curves

4x^{2} + 9y^{2} = 36

and

(2x)^{2} + (2y)^{2} = 31

. Then the square of the slope of the line L is ________________.

Q87·MathematicsSingle correctJEE Main 2021

For which of the following curves, the line

x+\sqrt{3}y=2\sqrt{3}

is the tangent at the point

\left(\frac{3\sqrt{3}}{2},\frac{1}{2}\right)

?

(A) $x^{2}+9y^{2}=9$
(B) $2x^{2}-18y^{2}=9$
(C) $y^{2}=\frac{1}{6\sqrt{3}}x$
(D) $x^{2}+y^{2}=7$

Q88·MathematicsSingle correctJEE Advanced 2020

Let a, b and

\lambda

be positive real numbers. Suppose P is an end point of the latus rectum of the parabola

y^{2} = 4\lambda x

, and suppose the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

(A) $\frac{1}{\sqrt{2}}$
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) $\frac{2}{5}$

Q89·MathematicsSingle correctJEE Main 2020

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies:

(A) $e^4 + 2e^2 - 1 = 0$
(B) $e^2 + e - 1 = 0$
(C) $e^2 + 2e - 1 = 0$
(D) $e^4 + e^2 - 1 = 0$

Q90·MathematicsSingle correctJEE Main 2020

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,

\dfrac{x^2}{4} + \dfrac{y^2}{2} = 1

from any of its foci?

(A) $(-2, \sqrt{3})$
(B) $(-1, \sqrt{2})$
(C) $(-1, \sqrt{3})$
(D) $(1, 2)$

Q91·MathematicsSingle correctJEE Main 2020

If the co-ordinates of two points A and B are

(\sqrt{7},0)

and

(-\sqrt{7},0)

respectively and P is any point on the conic

9x^{2}+16y^{2}=144

, then PA + PB is equal to:

(A)8
(B)16
(C)9
(D)6

Q92·MathematicsSingle correctJEE Main 2020

Let

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\,(a > b)

be given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function,

\phi(t) = \frac{5}{12} + t - t^{2}

, then

a^{2} + b^{2}

is equal to:

(A) $126$
(B) $135$
(C) $116$
(D) $145$

Q93·MathematicsSingle correctJEE Main 2020

Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is

\frac{1}{2}

. If P(1, β), β > 0 is a point on this ellipse, then the equation of the normal to it at P is:

(A)4x − 2y = 1
(B)4x − 3y = 2
(C)7x − 4y = 1
(D)8x − 2y = 5

Q94·MathematicsSingle correctJEE Main 2020

Let

e_1

and

e_2

be the eccentricities of the ellipse,

\frac{x^2}{25} + \frac{y^2}{b^2} = 1 \left( b < 5 \right)

and the hyperbola,

\frac{x^2}{16} - \frac{y^2}{b^2} = 1

respectively satisfying

e_1 e_2 = 1

. If

\alpha

and

\beta

are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair

\left( \alpha, \beta \right)

is equal to :

(A) $(8, 10)$
(B) $\left( \frac{24}{5}, 10 \right)$
(C) $\left( \frac{20}{3}, 12 \right)$
(D) $(8, 12)$

Q95·MathematicsSingle correctJEE Main 2020

The length of the minor axis (along y − axis) of an ellipse in the standard form is

\frac{4}{\sqrt{3}}

. If this ellipse touches the line,

x + 6y = 8

; then its eccentricity is:

(A) $\frac{1}{3}\sqrt{\frac{11}{3}}$
(B) $\frac{1}{2}\sqrt{\frac{5}{3}}$
(C) $\frac{1}{2}\sqrt{\frac{11}{3}}$
(D) $\sqrt{\frac{5}{6}}$

Q96·MathematicsSingle correctJEE Main 2020

If

e_1

and

e_2

are the eccentricities of the ellipse,

\dfrac{x^2}{18}+\dfrac{y^2}{4}=1

and the hyperbola,

\dfrac{x^2}{9}-\dfrac{y^2}{4}=1

respectively and

(e_1,e_2)

is a point on the ellipse,

15x^2+3y^2=k

, then k is equal to:

(A)16
(B)14
(C)17
(D)15

Q97·MathematicsSingle correctJEE Main 2020

Let the line

y=mx

and the ellipse

2x^{2}+y^{2}=1

intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at

\left(-\dfrac{1}{3\sqrt{2}},0\right)

and

(0,\beta)

then

\beta

is equal to:

(A) $\dfrac{2\sqrt{2}}{3}$
(B) $\dfrac{2}{3}$
(C) $\dfrac{2}{\sqrt{3}}$
(D) $\dfrac{\sqrt{2}}{3}$

Q98·MathematicsSingle correctJEE Main 2020

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is:

(A) $\sqrt{3}$
(B) $3\sqrt{2}$
(C) $\frac{3}{\sqrt{2}}$
(D) $2\sqrt{3}$

Q99·MathematicsSingle correctJEE Main 2020

If

3x+4y=12\sqrt{2}

is a tangent to the ellipse

\frac{x^2}{a^2}+\frac{y^2}{9}=1

for some

a \in R

, then the distance between the foci of the ellipse is:

(A) $2\sqrt{2}$
(B) $2\sqrt{7}$
(C) $4$
(D) $2\sqrt{5}$

Q100·MathematicsMultiple correctJEE Advanced 2019

Define the collections

\{E_1, E_2, E_3, .......\}

of ellipses and

\{R_1, R_2, R_3, ...\}

of rectangles as follows:

E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1

;

R_1

: rectangle of largest area, with sides parallel to the axes, inscribed in

E_1

;

E_n

: ellipse

\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1

of largest area inscribed in

R_{n-1}

,

n > 1

;

R_n

: rectangle of largest area, with sides parallel to the axes, inscribed in

E_n

,

n > 1

. Then which of the following options is/are correct?

(A) $\sum_{n=1}^{N} \left(\text{area of } R_n\right) < 24$ , for each positive integer N
(B)The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
(C)The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal
(D)The length of latus rectum of $E_9$ is $\frac{1}{6}$

Q101·MathematicsSingle correctJEE Main 2019

If the normal to the ellipse

3x^{2} + 4y^{2} = 12

at a point P on it is parallel to the line,

2x + y = 4

and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to :

(A) $\dfrac{\sqrt{157}}{2}$
(B) $\dfrac{5\sqrt{5}}{2}$
(C) $\dfrac{\sqrt{221}}{2}$
(D) $\dfrac{\sqrt{61}}{2}$

Q102·MathematicsSingle correctJEE Main 2019

An ellipse, with foci at (0, 2) and (0,

-2

) and minor axis of length 4, passes through which of the following points?

(A) $\left(2, \sqrt{2}\right)$
(B) $\left(2, 2\sqrt{2}\right)$
(C) $\left(1, 2\sqrt{2}\right)$
(D) $\left(\sqrt{2}, 2\right)$

Q103·MathematicsSingle correctJEE Main 2019

The tangent and normal to the ellipse

3x^{2} + 5y^{2} = 32

at the point

P(2, 2)

meet the x-axis at Q and R, respectively. Then the area(in sq. units) of the triangle PQR is

(A) $\dfrac{34}{15}$
(B) $\dfrac{68}{15}$
(C) $\dfrac{14}{3}$
(D) $\dfrac{16}{3}$

Q104·MathematicsSingle correctJEE Main 2019

If the tangents on the ellipse

4x^{2}+y^{2}=8

at the points (1, 2) and (a, b) are perpendicular to each other, then

a^{2}

is equal to:

(A) $\dfrac{2}{17}$
(B) $\dfrac{4}{17}$
(C) $\dfrac{64}{17}$
(D) $\dfrac{128}{17}$

Q105·MathematicsSingle correctJEE Main 2019

Let O (0, 0) and A (0, 1) be two fixed points. Then the locus of a point P such that the perimeter of

\triangle AOP

, is 4, is:

(A) $9x^{2}-8y^{2}+8y=16$
(B) $8x^{2}+9y^{2}-9y=18$
(C) $9x^{2}+8y^{2}-8y=16$
(D) $8x^{2}-9y^{2}+9y=18$

Q106·MathematicsSingle correctJEE Main 2019

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at

(0, 5\sqrt{3})

, then the length of its latus rectum is:

(A)6
(B)5
(C)8
(D)10

Q107·MathematicsSingle correctJEE Main 2019

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If

\triangle S'BS

is a right angled triangle with right angle at B and area

(\triangle S'BS) = 8

sq. units, then the length of a latus rectum of the ellipse is :

(A)4
(B) $2\sqrt{2}$
(C) $4\sqrt{2}$
(D)2

Q108·MathematicsSingle correctJEE Main 2019

If tangents are drawn to the ellipse

x^{2}+2y^{2}=2

at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curve:

(A) $\frac{1}{4x^{2}}+\frac{1}{2y^{2}}=1$
(B) $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$
(C) $\frac{1}{2x^{2}}+\frac{1}{4y^{2}}=1$
(D) $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$

Q109·MathematicsMultiple correctJEE Advanced 2018

Consider two straight lines, each of which is tangent to both the circle

x^{2} + y^{2} = \frac{1}{2}

and the parabola

y^{2} = 4x

. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is

\sqrt{2}

, then which of the following statement(s) is (are) TRUE ?

(A)For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is 1
(B)For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$
(C)The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{4\sqrt{2}}(\pi - 2)$
(D)The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{16}(\pi - 2)$

Q110·MathematicsSingle correctJEE Advanced 2017

Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. The tangent to a suitable conic (Column 1) at

\left(\sqrt{3}, \frac{1}{2}\right)

is found to be

\sqrt{3}x + 2y = 4

, then which of the following options is the only CORRECT combination ?

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $my = m^{2}x + a$	(P) $\left(\frac{a}{m^{2}}, \frac{2a}{m}\right)$
(II) $x^{2} + a^{2}y^{2} = a^{2}$	(ii) $y = mx + a\sqrt{m^{2} + 1}$	(Q) $\left(\frac{-ma}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}\right)$
(III) $y^{2} = 4ax$	(iii) $y = mx + \sqrt{a^{2}m^{2} - 1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} + 1}}, \frac{1}{\sqrt{a^{2}m^{2} + 1}}\right)$
(IV) $x^{2} - a^{2}y^{2} = a^{2}$	(iv) $y = mx + \sqrt{a^{2}m^{2} + 1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} - 1}}, \frac{-1}{\sqrt{a^{2}m^{2} - 1}}\right)$

(A)(II) (iii) (R)
(B)(IV) (iv) (S)
(C)(IV) (iii) (S)
(D)(II) (iv) (R)

Q111·MathematicsSingle correctJEE Advanced 2016

Let

F_{1}(x_{1}, 0)

and

F_{2}(x_{2}, 0)

, for

x_{1} < 0

and

x_{2} > 0

, be the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1

. Suppose a parabola having vertex at the origin and focus at

F_{2}

intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. The orthocentre of the triangle

F_{1}MN

is

(A) $\left( -\frac{9}{10}, 0 \right)$
(B) $\left( \frac{2}{3}, 0 \right)$
(C) $\left( \frac{9}{10}, 0 \right)$
(D) $\left( \frac{2}{3}, \sqrt{6} \right)$

Q112·MathematicsSingle correctJEE Advanced 2016

Let

F_{1}(x_{1}, 0)

and

F_{2}(x_{2}, 0)

, for

x_{1} < 0

and

x_{2} > 0

, be the foci of the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1

. Suppose a parabola having vertex at the origin and focus at

F_{2}

intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral

MF_{1}NF_{2}

is

(A)3 : 4
(B)4 : 5
(C)5 : 8
(D)2 : 3

Q113·MathematicsMultiple correctJEE Advanced 2015

Let

E_{1}

and

E_{2}

be two ellipses whose centers are at the origin. The major axes of

E_{1}

and

E_{2}

lie along the x-axis and the y-axis, respectively. Let

S

be the circle

x^{2} + (y - 1)^{2} = 2

. The straight line

x + y = 3

touches the curves

S

,

E_{1}

ad

E_{2}

at

P

,

Q

and

R

, respectively. Suppose that

PQ = PR = \dfrac{2\sqrt{2}}{3}

. If

e_{1}

and

e_{2}

are the eccentricities of

E_{1}

and

E_{2}

, respectively, then the correct expression(s) is(are)

(A) $e_{1}^{2} + e_{2}^{2} = \dfrac{43}{40}$
(B) $e_{1}e_{2} = \dfrac{\sqrt{7}}{2\sqrt{10}}$
(C) $\left|e_{1}^{2} - e_{2}^{2}\right| = \dfrac{5}{8}$
(D) $e_{1}e_{2} = \dfrac{\sqrt{3}}{4}$

Q114·MathematicsIntegerJEE Advanced 2013

A vertical line passing through the point

(h,0)

intersects the ellipse

\frac{x^2}{4}+\frac{y^2}{3}=1

at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If

\Delta(h)

= area of the triangle PQR,

\Delta_1=\max_{1/2\leq h\leq 1}\Delta(h)

and

\Delta_2=\min_{1/2\leq h\leq 1}\Delta(h)

, then

\frac{8}{\sqrt{5}}\Delta_1-8\Delta_2=

________

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