Question 1: Palette 1/24

JEE Advanced 2026 MCQ

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
(P) The circle with centre $(1,2)$ and touching the straight line $3x + 4y = 1$ passes through	(1) the point $(1,1)$
(Q) The common tangent to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ with positive slope, passes through	(2) the point $(7,9)$
(R) Let $M$ be the end point of the latus rectum of the ellipse $3x^2 + 4y^2 = 48$ such that $M$ lies in the first quadrant. Then the normal to the ellipse drawn at $M$ passes through	(3) the point $(3,2)$
(S) Let $H$ be the hyperbola whose centre is at the origin, one of the foci is at $(5,0)$, and one directrix is $5x + 16 = 0$ Then $H$ passes through	(4) the point $(2,5)$
	(5) the point $(8, 3\sqrt{3})$

A

(P) → (3), (Q) → (4), (R) → (1), (S) → (2)

B

(P) → (3), (Q) → (2), (R) → (1), (S) → (5)

C

(P) → (3), (Q) → (2), (R) → (4), (S) → (5)

D

(P) → (4), (Q) → (1), (R) → (2), (S) → (3)

Question 2: Palette 2/24

JEE Advanced 2020 MSQ

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 ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. 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

$\Delta = {b^4}$

Question 3: Palette 3/24

JEE Advanced 2018 MCQ

Let $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$, 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 $\Delta $LMN be $4\sqrt 3 $.

	List - I		List - II
P.	The length of the conjugate axis of H is	1.	8
Q.	The eccentricity of H is	2.	${4 \over {\sqrt 3 }}$
R.	The distance between the foci of H is	3.	${2 \over {\sqrt 3 }}$
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

Question 4: Palette 4/24

JEE Advanced 2018 MSQ

Let T be the line passing through the points P($-$2, 7) and Q(2, $-$5). Let F₁ be the set of al pairs of circles (S₁, S₂) such that T is tangent to S₁ at P and tangent to S₂ at Q, and also such that S₁ and S₂ touch each other at a point, say M. Let E₁ be the set representing the locus of M as the pair (S₁, S₂) varies in F₁. Let the set of all straight line segments joining a pair of distinct points of E₁ and passing through the point R(1, 1) be F₂. Let E₂ be the set of the mid-points of the line segments in the set F₂. Then, which of the following statement(s) is (are) TRUE?

A

The point ($-$2, 7) lies in E₁

B

The point $\left( {{4 \over 5},{7 \over 5}} \right)$ does not lie in E₂

C

The point $\left( {{1 \over 2},1} \right)$ lies in E₂

D

The point $\left( {0,{3 \over 2}} \right)$ does not lie in E₁

Question 5: Palette 5/24

JEE Advanced 2017 MCQ

For $a = \sqrt 2 $, if a tangent is drawn to a suitable conic (Column 1) at the point of contact ($-$1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

A

(I) (ii) Q)

B

(I) (ii) (P)

C

(III) (i) (P)

D

(II) (ii) (Q)

Question 6: Palette 6/24

JEE Advanced 2017 MCQ

The tangent to a suitable conic (Column 1) at $\left( {\sqrt 3 ,\,{1 \over 2}} \right)$ is found to be $\sqrt 3 x + 2y = 4$, then which of the following options is the only CORRECT combination?

A

(IV) (iv) (S)

B

(II) (iv) (R)

C

(IV) (iii) (S)

D

(II) (ii) (R)

Question 7: Palette 7/24

JEE Advanced 2017 MSQ

If $2x - y + 1 = 0$ is a tangent to the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1$ then which of the following CANNOT be sides of a right angled triangle?

A

a, 4, 1

B

2a, 4, 1

C

a, 4, 2

D

2a, 8, 1

Question 8: Palette 8/24

JEE Advanced 2015 MSQ

Consider the hyperbola $H:{x^2} - {y^2} = 1$ and a circle $S$ with center $N\left( {{x_2},0} \right)$. Suppose that $H$ and $S$ touch each other at a point $P\left( {{x_1},{y_1}} \right)$ 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 $PMN$, then the correct expressions(s) is(are)

A

${{dl} \over {d{x_1}}} = 1 - {1 \over {3x_1^2}}$ for ${x_1} > 1$

B

${{dm} \over {d{x_1}}} = {{{x_1}} \over {3\left( {\sqrt {x_1^2 - 1} } \right)}}$ for ${x_1} > 1$

C

${{dl} \over {d{x_1}}} = 1 + {1 \over {3x_1^2}}$ for ${x_1} > 1$

D

${{dm} \over {d{y_1}}} = {1 \over 3}$ for ${y_1} > 0$

Question 9: Palette 9/24

JEE Advanced 2012 MSQ

Tangents are drawn to the hyperbola ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1,$ parallel to the straight line $2x - y = 1,$ The points of contact of the tangents on the hyperbola are

A

$\left( {{9 \over {2\sqrt 2 }},{1 \over {\sqrt 2 }}} \right)$

B

$\left( -{{9 \over {2\sqrt 2 }},-{1 \over {\sqrt 2 }}} \right)$

C

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

D

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

Question 10: Palette 10/24

JEE Advanced 2011 MCQ

Let $P(6, 3)$ be a point on the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. If the normal at the point $P$ intersects the $x$-axis at $(9, 0)$, then the eccentricity of the hyperbola is

A

$\sqrt {{5 \over 2}} $

B

$\sqrt {{3 \over 2}} $

C

${\sqrt 2 }$

D

${\sqrt 3 }$

Question 11: Palette 11/24

JEE Advanced 2011 MSQ

Let the eccentricity of the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$ be reciprocal to that of the ellipse ${x^2} + 4{y^2} = 4$. If the hyperbola passes through a focus of the ellipse, then

A

the equation of the hyperbola is ${{{x^2}} \over 3} - {{{y^2}} \over 2} = 1$

B

a focus of the hyperbola is $(2, 0)$

C

theeccentricity of the hyperbola is $\sqrt {{5 \over 3}} $

D

The equation of the hyperbola is ${x^2} - 3{y^2} = 3$

Question 12: Palette 12/24

JEE Advanced 2010 MCQ

The circle ${x^2} + {y^2} - 8x = 0$ and hyperbola ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ intersect at the points $A$ and $B$.

Equation of the circle with $AB$ as its diameter is

A

${x^2} + {y^2} - 12x + 24 = 0$

B

${x^2} + {y^2} + 12x + 24 = 0$

C

${x^2} + {y^2} + 24x - 12 = 0$

D

${x^2} + {y^2} - 24x - 12 = 0$

Question 13: Palette 13/24

JEE Advanced 2010 MCQ

The circle ${x^2} + {y^2} - 8x = 0$ and hyperbola ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ intersect at the points $A$ and $B$.

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

A

$2x - \sqrt {5y} - 20 = 0$

B

$2x - \sqrt {5y} + 4 = 0$

C

$3x - 4y + 8 = 0$

D

$4x - 3y + 4 = 0$

Question 14: Palette 14/24

JEE Advanced 2008 MCQ

Consider a branch of the hyperbola $${x^2} - 2{y^2} - 2\sqrt 2 x - 4\sqrt 2 y - 6 = 0$$

with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $ABC$ is

A

$1 - \sqrt {{2 \over 3}} $

B

$\sqrt {{3 \over 2}} - 1$

C

$1 + \sqrt {{2 \over 3}} $

D

$\sqrt {{3 \over 2}} + 1$

Question 15: Palette 15/24

JEE Advanced 2007 MCQ

A hyperbola, having the transverse axis of length $2\sin \theta ,$ is confocal with the ellipse $3{x^2} + 4{y^2} = 12.$ Then its equation is

A

${x^2}\cos e{c^2}\theta - {y^2}{\sec ^2}\theta = 1$

B

${x^2}\cos e{c^2}\theta - {y^2}{\sec ^2}\theta = 1$

C

${x^2}{\sin ^2}\theta - {y^2}co{s^2}\theta = 1$

D

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

Question 16: Palette 16/24

JEE Advanced 2007 MCQ

A hyperbola, having the transverse axis of the length $2\sin \theta $, is confocal with the ellipse $3{x^2} + 4{y^2} = 12$. Then its equation is

A

${x^2}\cos e{c^2}\theta - {y^2}{\sec ^2}\theta = 1$

B

${x^2}{\sec ^2}\theta - {y^2}\cos e{c^2}\theta = 1$

C

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

D

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

Question 17: Palette 17/24

JEE Advanced 2006 MSQ

If a hyperbola passes through the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, and the product of eccentricities is 1 , then

A

the equation of hyperbola is $\frac{x^2}{9}-\frac{y^2}{16}=1$

B

the equation of hyperbola is $\frac{x^2}{9}-\frac{y^2}{25}=1$

C

focus of hyperbola is $(5,0)$

D

focus of hyperbola is $(5 \sqrt{3}, 0)$

Question 18: Palette 18/24

JEE Advanced 2005 MCQ

Tangents are drawn from any point on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ to the circle $x^{2}+y^{2}=9$. Find the locus of mid-point of the chord of contact.

A

${{{x^2}} \over 4} + {{{y^2}} \over 9} = {{{{({x^2} + {y^2})}^2}} \over {81}}$

B

${{{x^2}} \over 4} - {{{y^2}} \over 9} = {{{{({x^2} + {y^2})}^2}} \over {81}}$

C

${{{x^2}} \over 9} + {{{y^2}} \over 4} = {{{{({x^2} + {y^2})}^2}} \over {81}}$

D

${{{x^2}} \over 9} - {{{y^2}} \over 4} = {{{{({x^2} + {y^2})}^2}} \over {81}}$

Question 19: Palette 19/24

JEE Advanced 2004 MCQ

If the line $62x + \sqrt 6 y = 2$ touches the hyperbola ${x^2} - 2{y^2} = 4$, then the point of contact is

A

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

B

$\left( { - 5,\,2\sqrt 6 } \right)$

C

$\left( {{1 \over 2},{1 \over {\sqrt 6 }}} \right)$

D

$\left( {4, - \,\sqrt 6 } \right)$

Question 20: Palette 20/24

JEE Advanced 2003 MCQ

For hyperbola ${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$ which of the following remains constant with change in $'\alpha '$

A

abscissae of vertices

B

abscissae of foci

C

eccentricity

D

directrix

Question 21: Palette 21/24

JEE Advanced 1999 MCQ

Let $P$ $\left( {a\,\sec \,\theta ,\,\,b\,\tan \theta } \right)$ and $Q$ $\left( {a\,\sec \,\,\phi ,\,\,b\,\tan \,\phi } \right)$, where $\theta + \phi = \pi /2,$, be two points on the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$.

If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$, then $k$ is equal to

A

${{{a^2} + {b^2}} \over a}$

B

$ - \left( {{{{a^2} + {b^2}} \over a}} \right)$

C

${{{a^2} + {b^2}} \over b}$

D

$ - \left( {{{{a^2} + {b^2}} \over b}} \right)$

Question 22: Palette 22/24

JEE Advanced 1999 MCQ

If $x$ $=$ $9$ is the chord of contact of the hyperbola ${x^2} - {y^2} = 9,$ then the equation of the vcorresponding pair of tangents is

A

$9{x^2} - 8{y^2} + 18x - 9 = 0$

B

$9{x^2} - 8{y^2} - 18x + 9 = 0$

C

$9{x^2} - 8{y^2} - 18x - 9 = 0$

D

$9{x^2} - 8{y^2} + 18x + 9 = 0$

Question 23: Palette 23/24

JEE Advanced 1981 MCQ

The equation ${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1$ represents

A

an ellipse

B

a hyperbola

C

a circle

D

none of these

Question 24: Palette 24/24

JEE Advanced 1981 MCQ

Each of the four inequalties given below defines a region in the $xy$ plane. One of these four regions does not have the following property. For any two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ in the region, the point $\left( {{{{x_1} + {x_2}} \over 2},{{{y_1} + {y_2}} \over 2}} \right)$ is also in the region. The inequality defining this region is

A

${x^2} + 2{y^2} \le 1$

B

Max $\left\{ {\left| x \right|,\left| y \right|} \right\} \le 1$

C

${x^2} - {y^2} \le 1$

D

${y^2} - x \le 0$