How many questions from Hyperbola appear in JEE?

Hyperbola has appeared in 77 of the last 186 JEE Main and JEE Advanced papers — about 41% of them — contributing 80 questions in total across those papers.

Is Hyperbola an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 41% 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 Hyperbola 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.

Hyperbola JEE Previous Year Questions with Answers

Q1·MathematicsNumericalJEE Advanced 2026

Consider the ellipse

E

given by

\dfrac{x^{2}}{18} + \dfrac{y^{2}}{12} = 1

. Let

H

be the hyperbola whose eccentricity is the reciprocal of the eccentricity of

E

and whose foci are the same as that of

E

. Let

P

and

Q

be the points of intersection of

H

and the parabola

\sqrt{5}\, y = x^{2}

in the first quadrant. Let

d

be the distance between

P

and

Q

. If

a

and

b

are the integers such that

d^{2} = a + b\sqrt{5}

, then the value of

a - b

is _____.

Q2·MathematicsSingle correctJEE Main 2026

The eccentricity of an ellipse

E

with centre at the origin

O

is

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

and its directrices are

x = \pm\frac{4\sqrt{6}}{3}

. Let

H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

be a hyperbola whose eccentricity is equal to the length of semi-major axis of

E

, and whose length of latus rectum is equal to the length of minor axis of

E

. Then the distance between the foci of

H

is :

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

Q3·MathematicsSingle correctJEE Main 2026

If the eccentricity e of the hyperbola

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

, passing through (6, 4√3), satisfies 15(e

^2

+ 1) = 34e, then the length of the latus rectum of the hyperbola

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

is:

(A)10
(B)20
(C)25
(D)30

Q4·MathematicsSingle correctJEE Main 2026

Let the eccentricity e of a hyperbola satisfy the equation

6e^2 - 11e + 3 = 0

. If the foci of the hyperbola are (3, 5) and (3, −4), then the length of its latus rectum is :

(A) $\frac{11}{3}$
(B) $\frac{17}{3}$
(C) $\frac{15}{2}$
(D) $\frac{17}{2}$

Q5·MathematicsSingle correctJEE Main 2026

Let

H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is

\frac{8}{3}

. If the line

x = \alpha

intersects the hyperbola

H

at the points

A

and

B

such that the area of the triangle

AOB

is

4\sqrt{15}

, where

O

is the origin, then

\alpha^{2}

equals

(A)12
(B)16
(C)24
(D)25

Q6·MathematicsSingle correctJEE Main 2026

Let O be the origin, and P and Q be two points on the rectangular hyperbola

xy = 12

such that the mid point of the line segment PQ is

\left(\frac{1}{2}, -\frac{1}{2}\right)

. Then the area of the triangle OPQ equals:

(A) $\frac{3}{2}$
(B) $\frac{5}{2}$
(C) $\frac{7}{2}$
(D) $\frac{9}{2}$

Q7·MathematicsSingle correctJEE Main 2026

Let the ellipse

E : \dfrac{x^{2}}{144} + \dfrac{y^{2}}{169} = 1

and the hyperbola

H : \dfrac{x^{2}}{16} - \dfrac{y^{2}}{\lambda^{2}} = -1

have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H, then the value of 24(e + L) is :

(A)296
(B)126
(C)148
(D)67

Q8·MathematicsNumericalJEE Main 2026

For some

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

, let the eccentricity and the length of the latus rectum of the hyperbola

x^{2} - y^{2}\sec^{2}\theta = 8

be

e_1

and

\ell_1

, respectively, and let the eccentricity and the length of the latus rectum of the ellipse

x^{2}\sec^{2}\theta + y^{2} = 6

be

e_2

and

\ell_2

, respectively. If

e_1^{2} = e_2^{2}\left(\sec^{2}\theta + 1\right)

, then

\left(\dfrac{\ell_1\ell_2}{e_1e_2}\right)\tan^{2}\theta

is equal to _____.

Q9·MathematicsSingle correctJEE Main 2026

Let PQ be a chord of the hyperbola

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

, perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is

\sqrt{3}

, then the area of the triangle OPQ is :

(A) $2\sqrt{3}$
(B) $\dfrac{8\sqrt{3}}{5}$
(C) $\dfrac{11}{5}$
(D) $\dfrac{9}{5}$

Q10·MathematicsSingle correctJEE Main 2026

Let the domain of the function

f(x) = \log_3\log_5\log_7 (9x - x^2 - 13)

be the interval (m, n). Let the hyperbola

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

have eccentricity

\frac{n}{3}

and the length of the latus rectum

\frac{8m}{3}

. Then

b^2 - a^2

is equal to :

(A)5
(B)11
(C)9
(D)7

Q11·MathematicsSingle correctJEE Main 2026

Let P

\left(10,\ 2\sqrt{15}\right)

be a point on the hyperbola

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

, whose foci are S and S'. If the length of its latus rectum is 8, then the square of the area of

\Delta PSS'

is equal to :

(A)4200
(B)900
(C)1462
(D)2700

Q12·MathematicsSingle correctJEE Main 2026

If the line αx + 2y = 1, where α ∈ ℝ, does not meet the hyperbola

x^2 - 9y^2 = 9

, then a possible value of α is :

(A)0.6
(B)0.8
(C)0.5
(D)0.7

Q13·MathematicsSingle correctJEE Main 2026

Let the foci of hyperbola coincide with the foci of the ellipse

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

. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:

(A)12
(B)16
(C) $\frac{96}{\sqrt{5}}$
(D) $24\sqrt{5}$

Q14·MathematicsSingle correctJEE Advanced 2025

Let S denote the locus of the point of intersection of the pair of lines

4x - 3y = 12\alpha

,

4\alpha x + 3\alpha y = 12

, where α varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points (p, 0) and (0, q), q > 0, and parallel to the line

4x - \frac{3}{\sqrt{2}}y = 0

. Then the value of pq is

(A) $-6\sqrt{2}$
(B) $-3\sqrt{2}$
(C) $-9\sqrt{2}$
(D) $-12\sqrt{2}$

Q15·MathematicsIntegerJEE Main 2025

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be

2a

and

2b

, respectively, and one focus and the corresponding directrix of this hyperbola be

(-5,0)

and

5x+9=0

, respectively. If the product of the focal distances of a point

\left(\alpha,2\sqrt{5}\right)

on the hyperbola is

p

, then

4p

is equal to __________.

Q16·MathematicsIntegerJEE Main 2025

Consider the hyperbola

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

having one of its focus at

P(-3,0)

. If the latus rectum through its other focus subtends a right angle at

P

and

a^2b^2=\alpha\sqrt{2}-\beta

,

\alpha,\beta\in\mathbb{N}

. Then

\alpha+\beta=

______.

Q17·MathematicsSingle correctJEE Main 2025

Let the sum of the focal distances of the point

P(4,3)

on the hyperbola

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

be

8\sqrt{\dfrac{5}{3}}

. If for

H

, the length of the latus rectum is

l

and the product of the focal distances of the point

P

is

m

, then

9l^2+6m

is equal to:

(A)184
(B)186
(C)185
(D)187

Q18·MathematicsIntegerJEE Main 2025

Let the product of the focal distances of the point

P(4,2\sqrt3)

on the hyperbola

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

be 32. Let the length of the conjugate axis of

H

be

p

and the length of its latus rectum be

q

. Then

p^2+q^2

is equal to __________.

Q19·MathematicsIntegerJEE Main 2025

If the equation of the hyperbola with foci

(4,2)

and

(8,2)

is

3x^2-y^2-\alpha x+\beta y+\gamma=0

, then

\alpha+\beta+\gamma

is equal to __________.

Q20·MathematicsSingle correctJEE Main 2025

If A and B are the points of intersection of the circle

x^2+y^2-8x=0

and the hyperbola

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

and a point P moves on the line

2x-3y+4=0

, then the centroid of

\triangle PAB

lies on the line:

(A) $4x-9y=12$
(B) $x+9y=36$
(C) $9x-9y=32$
(D) $6x-9y=20$

Q21·MathematicsIntegerJEE Main 2025

Let

H_1:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1

and

H_2:-\dfrac{x^2}{A^2}+\dfrac{y^2}{B^2}=1

be two hyperbolas having length of latus rectums

15\sqrt2

and

12\sqrt5

respectively. Let their eccentricities be

e_1=\sqrt{\dfrac{5}{2}}

and

e_2

respectively. If the product of the lengths of their transverse axes is

100\sqrt{10}

, then

25e_2^2

is equal to __________.

Q22·MathematicsSingle correctJEE Main 2025

The foci of a hyperbola are

(1,14)

and

(1,-12)

, and it passes through

(1,6)

. The length of its latus rectum is:

(A) $\tfrac{25}{6}$
(B) $\tfrac{24}{5}$
(C) $\tfrac{288}{5}$
(D) $\tfrac{144}{5}$

Q23·MathematicsSingle correctJEE Main 2024

Let the foci of a hyperbola H coincide with the foci of the ellipse

E:\dfrac{(x-1)^{2}}{100}+\dfrac{(y-1)^{2}}{75}=1

and the eccentricity of the hyperbola H be the reciprocal of the eccentricity of the ellipse E. If the length of the transverse axis of H is

\alpha

and the length of its conjugate axis is

\beta

, then

3\alpha^{2}+2\beta^{2}

is equal to:

(A) $242$
(B) $225$
(C) $237$
(D) $205$

Q24·MathematicsSingle correctJEE Main 2024

Let

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

be the hyperbola, whose eccentricity is

\sqrt3

and the length of the latus rectum is

4\sqrt3

. Suppose the point

(\alpha,6)

,

\alpha>0

lies on

H

. If

\beta

is the product of the focal distances of the point

(\alpha,6)

, then

\alpha^2+\beta

is equal to:

(A) $170$
(B) $171$
(C) $169$
(D) $172$

Q25·MathematicsNumericalJEE Main 2024

Let S be the focus of the hyperbola

\dfrac{x^2}{3}-\dfrac{y^2}{5}=1

, on the positive x-axis. Let C be the circle with its centre at

A(\sqrt{6},\sqrt{5})

and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to _______ .

Q26·MathematicsNumericalJEE Main 2024

The length of the latus rectum and directrices of a hyperbola with eccentricity

e

are

9

and

x=\pm\frac{4}{\sqrt{3}}

, respectively. Let the line

y-\sqrt{3}x+\sqrt{3}=0

touch this hyperbola at

(x_{0},y_{0})

. If

m

is the product of the focal distances of the point

(x_{0},y_{0})

, then

4e^{2}+m

is equal to ______.

Q27·MathematicsSingle correctJEE Main 2024

Consider the hyperbola H having centre at the origin and foci on the x-axis. Let

C_1

be the circle touching the hyperbola H and having the centre at the origin. Let

C_2

be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq. units) of

C_1

and

C_2

are

36\pi

and

4\pi

, respectively, then the length (in units) of latus rectum of H is

(A) $\tfrac{28}{3}$
(B) $\tfrac{14}{3}$
(C) $\tfrac{10}{3}$
(D) $\tfrac{11}{3}$

Q28·MathematicsSingle correctJEE Main 2024

Let

P

be a point on the hyperbola

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

, in the first quadrant such that the area of triangle formed by

P

and the two foci of

H

is

2\sqrt{13}

. Then, the square of the distance of

P

from the origin is:

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

Q29·MathematicsNumericalJEE Main 2024

Let the latus rectum of the hyperbola

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

subtend an angle of

\dfrac\pi3

at the centre of the hyperbola. If

b^2

is equal to

\dfrac{l}{m}(1+\sqrt n)

, where

l

and

m

are co-prime numbers, then

l^2+m^2+n^2

is equal to ___

Q30·MathematicsNumericalJEE Main 2023

Let

m_{1}

and

m_{2}

be the slopes of the tangents drawn from the point

P(4,1)

to the hyperbola

H:\dfrac{y^{2}}{25}-\dfrac{x^{2}}{16}=1

. If

Q

is the point from which the tangents drawn to

H

have slopes

|m_{1}|

and

|m_{2}|

and they make positive intercepts

\alpha

and

\beta

on the

x

-axis, then

\dfrac{(PQ)^{2}}{\alpha\beta}

is equal to _____.

Q31·MathematicsNumericalJEE Main 2023

The foci of a hyperbola are

(\pm 2, 0)

and its eccentricity is

\dfrac{3}{2}

. A tangent, perpendicular to the line

2x + 3y = 6

, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x- and y-axes are

a

and

b

respectively, then

|6a| + |5b|

is equal to

Q32·MathematicsNumericalJEE Main 2023

Let the tangent to the parabola

y^2=12x

at the point

(3,\alpha)

be perpendicular to the line

2x+2y=3

. Then the square of the distance of the point

(6,-4)

from the normal to the hyperbola

\alpha^2 x^2-9y^2=9\alpha^2

at its point

(\alpha-1,\alpha+2)

is equal to

Q33·MathematicsNumericalJEE Main 2023

Let

H_n = \frac{x^2}{1 + n} - \frac{y^2}{3 + n} = 1

,

n \in \mathbb{N}

. Let

k

be the smallest even value of

n

such that the eccentricity of

H_k

is a rational number. If

l

is length of the latus rectum of

H_k

, then

21l

is equal to _______ .

Q34·MathematicsSingle correctJEE Main 2023

Let

P(x_0,y_0)

be the point on the hyperbola

3x^2-4y^2=36

, which is nearest to the line

3x+2y=1

. Then

\sqrt2(y_0-x_0)

is equal to:

(A) $-9$
(B) $-3$
(C) $3$
(D) $9$

Q35·MathematicsSingle correctJEE Main 2023

Let H be the hyperbola, whose foci are

(1\pm\sqrt2,0)

and eccentricity is

\sqrt2

. Then the length of its latus rectum is

(A) $\dfrac32$
(B) $2$
(C) $3$
(D) $\dfrac52$

Q36·MathematicsNumericalJEE Main 2023

The vertices of a hyperbola

H

are

(\pm 6,0)

and its eccentricity is

\dfrac{\sqrt{5}}{2}

. Let

N

be the normal to

H

at a point in the first quadrant and parallel to the line

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

. If

d

is the length of the line segment of

N

between

H

and the

y

-axis, then

d^2

is equal to _______.

Q37·MathematicsSingle correctJEE Main 2023

Let

T

and

C

respectively be the transverse and conjugate axes of the hyperbola

16x^2 - y^2 + 64x + 4y + 44 = 0

. Then the area of the region above the parabola

x^2 = y + 4

, below the transverse axis

T

and on the right of the conjugate axis

C

is:

(A) $4\sqrt{6} + \dfrac{28}{3}$
(B) $4\sqrt{6} - \dfrac{44}{3}$
(C) $4\sqrt{6} + \dfrac{44}{3}$
(D) $4\sqrt{6} - \dfrac{28}{3}$

Q38·MathematicsIntegerJEE Advanced 2022

Consider the hyperbola

\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1

with foci at S and S

_{1}

, where S lies on the positive x-axis. Let P be a point on the hyperbola, in the first quadrant. Let

\angle SPS_{1} = \alpha

, with

\alpha < \frac{\pi}{2}

. The straight line passing through the point S and having the same slope as that of the tangent at P to the hyperbola, intersects the straight line S

_{1}

P at P

_{1}

. Let δ be the distance of P from the straight line SP

_{1}

, and β = S

_{1}

P. Then the greatest integer less than or equal to

\frac{\beta\delta}{9}\sin\frac{\alpha}{2}

is ________.

Q39·MathematicsSingle correctJEE Main 2022

Let the hyperbola

H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1

pass through the point

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

. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

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

Q40·MathematicsNumericalJEE Main 2022

A common tangent T to the curves

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

and

C_{2} : \frac{x^{2}}{42} - \frac{y^{2}}{143} = 1

does not pass through the fourth quadrant. If T touches

C_{1}

at

(x_{1}, y_{1})

and

C_{2}

at

(x_{2}, y_{2})

, then

|2x_{1} + x_{2}|

is equal to ________.

Q41·MathematicsSingle correctJEE Main 2022

If the line

x-1 = 0

, is a directrix of the hyperbola

kx^{2}-y^{2}=6

, then the hyperbola passes through the point

(A) $\left(-2\sqrt{5},6\right)$
(B) $\left(-\sqrt{5},3\right)$
(C) $\left(\sqrt{5},-2\right)$
(D) $\left(2\sqrt{5},3\sqrt{6}\right)$

Q42·MathematicsSingle correctJEE Main 2022

Let the tangent drawn to the parabola

y^{2} = 24x

at the point

(\alpha, \beta)

is perpendicular to the line 2x +2y = 5. Then the normal to the hyperbola

\dfrac{x^{2}}{\alpha^{2}} - \dfrac{y^{2}}{\beta^{2}} = 1

at the point

(\alpha+4,\beta+4)

does NOT pass through the point :

(A)(25, 10)
(B)(20, 12)
(C)(30, 8)
(D)(15, 13)

Q43·MathematicsNumericalJEE Main 2022

Let the equation of two diameters of a circle

x^{2} + y^{2} - 2x + 2fy + 1 = 0

be

2px - y = 1

and

2x + py = 4p

. Then the slope

m \in (0, \infty)

of the tangent to the hyperbola

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

passing through the centre of the circle is equal to ________.

Q44·MathematicsNumericalJEE Main 2022

Let

H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

, a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is

4\left(2\sqrt{2} + \sqrt{14}\right)

. If the eccentricity H is

\frac{\sqrt{11}}{2}

, then value of

a^2 + b^2

is equal to __________.

Q45·MathematicsSingle correctJEE Main 2022

Let the eccentricity of the hyperbola

\mathrm{H}:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1

be

\sqrt{\frac{5}{2}}

and length of its latus rectum be

6\sqrt{2},

If

y = 2x + c

is a tangent to the hyperbola H, then the value of

c^{2}

is equal to

(A)18
(B)20
(C)24
(D)32

Q46·MathematicsSingle correctJEE Main 2022

Let

a > 0

,

b > 0

. Let e and

\ell

respectively be the eccentricity and length of the latus rectum of the hyperbola

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

. Let

e'

and

\ell'

respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If

e^{2} = \frac{11}{14}\ell

and

\left(e'\right)^{2} = \frac{11}{8}\ell'

, then the value of

77a + 44b

is equal to

(A)100
(B)110
(C)120
(D)130

Q47·MathematicsNumericalJEE Main 2022

Let a line

L_{1}

be tangent to the hyperbola

\frac{x^{2}}{16}-\frac{y^{2}}{4}=1

and let

L_{2}

be the line passing through the origin and perpendicular to

L_{1}

. If the locus of the point of intersection of

L_{1}

and

L_{2}

is

(x^{2}+y^{2})^{2}=\alpha x^{2}+\beta y^{2}

, then

\alpha+\beta

is equal to ______ .

Q48·MathematicsSingle correctJEE Main 2022

The normal to the hyperbola

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

at the point

\left(8,3\sqrt{3}\right)

on it passes through the point :

(A) $\left(15,-2\sqrt{3}\right)$
(B) $\left(9,2\sqrt{3}\right)$
(C) $\left(-1,9\sqrt{3}\right)$
(D) $\left(-1,6\sqrt{3}\right)$

Q49·MathematicsNumericalJEE Main 2022

Let the eccentricity of the hyperbola

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

be

\frac{5}{4}

. If the equation of the normal at the point

\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)

on the hyperbola is

8\sqrt{5}x + \beta y = \lambda

, then

\lambda - \beta

is equal to

Q50·MathematicsNumericalJEE Main 2022

Let the hyperbola

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

and the ellipse

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

be such that the length of latus rectum of H is equal to the length of latus rectum of E. If

e_{H}

and

e_{E}

are the eccentricities of H and E respectively, then the value of

12\left(e_{H}^{2} + e_{E}^{2}\right)

is equal to ______.

Q51·MathematicsSingle correctJEE Main 2022

Let

\lambda x-2y=\mu

be a tangent to the hyperbola

a^{2}x^{2}-y^{2}=b^{2}

. Then

\left(\dfrac{\lambda}{a}\right)^{2}-\left(\dfrac{\mu}{b}\right)^{2}

is equal to:

(A) $-2$
(B) $-4$
(C)2
(D)4

Q52·MathematicsNumericalJEE Advanced 2021

Consider the lines

L_1

and

L_2

defined by

L_1 : x\sqrt{2} + y - 1 = 0

and

L_2 : x\sqrt{2} - y + 1 = 0

For a fixed constant

\lambda

, let C be the locus of a point P such that the product of the distance of P from

L_1

and the distance of P from

L_2

is

\lambda^2

. The line

y = 2x + 1

meets C at two points R and S, where the distance between R and S is

\sqrt{270}

. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'. The value of

\lambda^2

is ________.

Q53·MathematicsNumericalJEE Advanced 2021

Consider the lines

L_1

and

L_2

defined by

L_1 : x\sqrt{2} + y - 1 = 0

and

L_2 : x\sqrt{2} - y + 1 = 0

For a fixed constant

\lambda

, let C be the locus of a point P such that the product of the distance of P from

L_1

and the distance of P from

L_2

is

\lambda^2

. The line

y = 2x + 1

meets C at two points R and S, where the distance between R and S is

\sqrt{270}

. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'. The value of D is ________.

Q54·MathematicsSingle correctJEE Main 2021

The point

P\left(-2\sqrt{6}, \sqrt{3}\right)

lies on the hyperbola

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

having eccentricity

\frac{\sqrt{5}}{2}

. If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to :

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

Q55·MathematicsSingle correctJEE Main 2021

The locus of the mid points of the chords of the hyperbola

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

, which touch the parabola

y^{2} = 8x

, is :

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

Q56·MathematicsSingle correctJEE Main 2021

The locus of the centroid of the triangle formed by any point P on this hyperbola

16x^{2}-9y^{2}+32x+36y-164=0

, and its foci is :

(A) $16x^{2}-9y^{2}+32x+36y-144=0$
(B) $16x^{2}-9y^{2}+32x+36y-36=0$
(C) $9x^{2}-16y^{2}+36x+32y-144=0$
(D) $9x^{2}-16y^{2}+36x+32y-36=0$

Q57·MathematicsSingle correctJEE Main 2021

Let a line L :

2x + y = k

,

k > 0

be a tangent to the hyperbola

x^2 - y^2 = 3

. If L is also a tangent to the parabola

y^2 = \alpha x

, then

\alpha

is equal to :

(A) $24$
(B) $-24$
(C) $-12$
(D) $12$

Q58·MathematicsNumericalJEE Main 2021

A square ABCD has all its vertices on the curve

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

. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is

Q59·MathematicsSingle correctJEE Main 2021

Consider a hyperbola H :

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

. Let the tangent at a point

P\left(4, \sqrt{6}\right)

meet the x-axis at Q and latus rectum at

R(x_{1}, y_{1})

,

x_{1} > 0

. If F is a focus of H which is nearer to the point P, then the area of ΔQFR is equal to

(A) $4\sqrt{6}$
(B) $\sqrt{6}-1$
(C) $\frac{7}{\sqrt{6}}-2$
(D) $4\sqrt{6}-1$

Q60·MathematicsSingle correctJEE Main 2021

The locus of the midpoints of the chord of the circle,

x^{2} + y^{2} = 25

which is tangent to the hyperbola,

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

is :

(A) $(x^{2} + y^{2})^{2} - 16x^{2} + 9y^{2} = 0$
(B) $(x^{2} + y^{2})^{2} - 9x^{2} + 144y^{2} = 0$
(C) $(x^{2} + y^{2})^{2} - 9x^{2} - 16y^{2} = 0$
(D) $(x^{2} + y^{2})^{2} - 9x^{2} + 16y^{2} = 0$

Q61·MathematicsSingle correctJEE Main 2021

A hyperbola passes through the foci of the ellipse x

^{2}

+y

^{2}

=1 and its transverse and conjugate 2516 axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

(A)x $^{2}$ –y $^{2}$ =1 94
(B)x $^{2}$ –y $^{2}$ =1 916
(C)x $^{2}$ – y $^{2}$ = 9
(D)x $^{2}$ –y $^{2}$ =1 925

Q62·MathematicsNumericalJEE Main 2021

The locus of the point of intersection of the lines

(\sqrt{3})kx + ky - 4\sqrt{3} = 0

and

\sqrt{3}x - y - 4(\sqrt{3})k = 0

is a conic, whose eccentricity is __________.

Q63·MathematicsMultiple correctJEE Advanced 2020

Let

a

and

b

be positive real numbers such that

a > 1

and

b < a

. Let

P

be a point in the first quadrant that lies on the hyperbola

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

. Suppose the tangent to the hyperbola at

P

passes through the point

(1,0)

, and suppose the normal to the hyperbola at

P

cuts off equal intercepts on the coordinate axes. Let

\Delta

denote the area of the triangle formed by the tangent at

P

, the normal at

P

and the

x

-axis. If

e

denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

(A) $1 < e < \sqrt{2}$
(B) $\sqrt{2} < e < 2$
(C) $\Delta = a^{4}$
(D) $\Delta = b^{4}$

Q64·MathematicsSingle correctJEE Main 2020

If the line y = mx + c is a common tangent to the hyperbola

\dfrac{x^2}{100} - \dfrac{y^2}{64} = 1

and the circle

x^2 + y^2 = 36

, then which one of the following is true?

(A) $4c^2 = 369$
(B) $c^2 = 369$
(C) $5m = 4$
(D) $8m + 5 = 0$

Q65·MathematicsSingle correctJEE Main 2020

Let P (3, 3) be a point on the hyperbola,

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

. If the normal to it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair

(a^{2}, e^{2})

is equal to:

(A) $\left( \frac{3}{2},\, 2 \right)$
(B) $\left( \frac{9}{2},\, 2 \right)$
(C) $(9,\, 3)$
(D) $\left( \frac{9}{2},\, 3 \right)$

Q66·MathematicsSingle correctJEE Main 2020

A hyperbola having the transverse axis of length

\sqrt{2}

has the same foci as that of the ellipse

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

, then this hyperbola does not pass through which of the following points?

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

Q67·MathematicsSingle correctJEE Main 2020

If a hyperbola passes through the point P (10, 16) and it has vertices at

(\pm 6, 0)

then the equation of the normal to it at P is:

(A) $2x+5y=100$
(B) $x+3y=58$
(C) $3x+4y=94$
(D) $x+2y=42$

Q68·MathematicsSingle correctJEE Main 2019

Let P be the point of intersection of the common tangents to the parabola

y^{2}=12x

and the hyperbola

8x^{2}-y^{2}=8

. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio:

(A)2:1
(B)13:11
(C)5:4
(D)14:13

Q69·MathematicsSingle correctJEE Main 2019

If a directrix of a hyperbola centered at the origin and passing through the point

(4,-2\sqrt{3})

is

5x=4\sqrt{5}

and its eccentricity is e, then

(A) $4e^{4}+8e^{2}-35=0$
(B) $4e^{4}-24e^{2}+35=0$
(C) $4e^{4}-12e^{2}-27=0$
(D) $4e^{4}-24e^{2}+27=0$

Q70·MathematicsSingle correctJEE Main 2019

If 5x + 9 = 0 is the directrix of the hyperbola

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

, then its corresponding focus is

(A)(5, 0)
(B) $\left(\dfrac{5}{3}, 0\right)$
(C)(-5, 0)
(D) $\left(-\dfrac{5}{3}, 0\right)$

Q71·MathematicsSingle correctJEE Main 2019

If the line

y=mx+7\sqrt{3}

is normal to the hyperbola

\dfrac{x^{2}}{24}-\dfrac{y^{2}}{18}=1

, then a value of m is:

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

Q72·MathematicsSingle correctJEE Main 2019

If the eccentricity of the standard hyperbola passing, through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:

(A) $2x-3y+10=0$
(B) $x-2y+8=0$
(C) $2x-y-2=0$
(D) $3x-2y=0$

Q73·MathematicsSingle correctJEE Main 2019

If the vertices of a hyperbola be at

(-2, 0)

and

(2, 0)

and one of its foci be at

(-3, 0)

, then which one of the following points does not lie on this hyperbola?

(A) $\left(-6,2\sqrt{10}\right)$
(B) $\left(2\sqrt{6},5\right)$
(C) $\left(4,\sqrt{15}\right)$
(D) $\left(6,5\sqrt{2}\right)$

Q74·MathematicsSingle correctJEE Main 2019

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:

(A) $\dfrac{13}{12}$
(B)2
(C) $\dfrac{13}{6}$
(D) $\dfrac{13}{8}$

Q75·MathematicsSingle correctJEE Main 2019

The equation of a tangent to the hyperbola

4x^2 - 5y^2 = 20

parallel to the line

x - y = 2

is:

(A)x − y + 1 = 0
(B)x − y + 7 = 0
(C)x − y + 9 = 0
(D)x − y − 3 = 0

Q76·MathematicsSingle correctJEE Main 2019

Let

0 < \theta < \dfrac{\pi}{2}

. If the eccentricity of the hyperbola

\dfrac{x^2}{\cos^2\theta} - \dfrac{y^2}{\sin^2\theta} = 1

is greater than 2, then the length of its latus rectum lies in the interval:

(A) $(3, \infty)$
(B) $\left(\dfrac{3}{2}, 2\right]$
(C) $(2, 3]$
(D) $\left(1, \dfrac{3}{2}\right]$

Q77·MathematicsSingle correctJEE Main 2019

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x - axis. Then the eccentricity of the hyperbola is:

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

Q78·MathematicsSingle correctJEE Advanced 2018

Let H :

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

, where

a > b > 0

, be a hyperbola in the xy-plane whose conjugate axis LM subtends an angle of

60^{\circ}

at one of its vertices N. Let the area of the triangle LMN be

4\sqrt{3}

. The correct option is :

LIST-I	LIST-II
P.The length of the conjugate axis of H is	1.8
Q.The eccentricity of H is	2. $\frac{4}{\sqrt{3}}$
R.The distance between the foci of H is	3. $\frac{2}{\sqrt{3}}$
S.The length of the latus rectum of H is	4.4

(A)P $\to$ 4; Q $\to$ 2; R $\to$ 1; S $\to$ 3
(B)P $\to$ 4; Q $\to$ 3; R $\to$ 1; S $\to$ 2
(C)P $\to$ 4; Q $\to$ 1; R $\to$ 3; S $\to$ 2
(D)P $\to$ 3; Q $\to$ 4; R $\to$ 2; S $\to$ 1

Q79·MathematicsMultiple correctJEE Advanced 2017

If

2x - y + 1 = 0

is a tangent to the hyperbola

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

, then which of the following CANNOT be sides of a right angled triangle ?

(A) $2a$ , $4$ , $1$
(B) $2a$ , $8$ , $1$
(C) $a$ , $4$ , $1$
(D) $a$ , $4$ , $2$

Q80·MathematicsMultiple correctJEE Advanced 2015

Consider the hyperbola

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

and a circle S with center

N(x_{2}, 0)

. Suppose that H and S touch each other at a point

P(x_{1}, y_{1})

with

x_{1} > 1

and

y_{1} > 0

. The common tangent to H and S at P intersects the x-axis at point M. If

(l, m)

is the centroid of the triangle

\Delta PMN

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

(A) $\dfrac{dl}{dx_{1}} = 1 - \dfrac{1}{3x_{1}^{2}}$ for $x_{1} > 1$
(B) $\dfrac{dm}{dx_{1}} = \dfrac{x_{1}}{3\left(\sqrt{x_{1}^{2} - 1}\right)}$ for $x_{1} > 1$
(C) $\dfrac{dl}{dx_{1}} = 1 + \dfrac{1}{3x_{1}^{2}}$ for $x_{1} > 1$
(D) $\dfrac{dm}{dy_{1}} = \dfrac{1}{3}$ for $y_{1} > 0$

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