Question 1: Palette 1/32

JEE Advanced 2025 MSQ

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ 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 $\angle R O M=\frac{\pi}{6}$ and $\angle S O M=\frac{\pi}{3}$, where $O$ denotes the origin $(0,0)$. Let $|X Y|$ denote the length of the line segment $X Y$.

Then which of the following statements is (are) TRUE?

A

The equation of the line joining P and Q is $2x + 3y = 3(1 + \sqrt{3})$

B

The equation of the line joining P and Q is $2x + y = 3(1 + \sqrt{3})$

C

If $N_2 = (x_2, 0)$, then $3|N_2Q| = 2|N_2S|$

D

If $N_1 = (x_1, 0)$, then $9|N_1P| = 4|N_1R|$

Question 2: Palette 2/32

JEE Advanced 2024 MCQ

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 center of the ellipse. If the area of the triangle $\triangle O R T$ 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

$q=1, p=6 \sqrt{3}$

Question 3: Palette 3/32

JEE Advanced 2023 MSQ

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 $P: y^2=12 x$. 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 $A_1 A_2 A_3 A_4$ is 35 square units

B

The area of the quadrilateral $A_1 A_2 A_3 A_4$ 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 $(-6,0)$

Question 4: Palette 4/32

JEE Advanced 2022 MCQ

Consider the ellipse

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

Let $H(\alpha, 0), 0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent 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.

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

The correct option is:

A

$(\mathrm{I}) \rightarrow(\mathrm{R}) ;(\mathrm{II}) \rightarrow(\mathrm{S}) ;(\mathrm{III}) \rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$

B

(I) $\rightarrow$ (R); (II) $\rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$

C

(I) $\rightarrow(\mathrm{Q}) ;(\mathrm{II}) \rightarrow(\mathrm{T}) ;(\mathrm{III}) \rightarrow(\mathrm{S}) ;(\mathrm{IV}) \rightarrow(\mathrm{P})$

D

(I) $\rightarrow$ (Q); (II) $\rightarrow$ (S); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (P)

Question 5: Palette 5/32

JEE Advanced 2019 MSQ

Define the collections {E₁, E₂, E₃, ...} of ellipses and {R₁, R₂, R₃.....} of rectangles as follows :

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

R₁ : rectangle of largest area, with sides parallel to the axes, inscribed in E₁;

En : ellipse ${{{x^2}} \over {a_n^2}} + {{{y^2}} \over {b_n^2}} = 1$ of the 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

The eccentricities of E₁₈ and E₁₉ are not equal.

B

The distance of a focus from the centre in E₉ is ${{\sqrt 5 } \over {32}}$.

C

$\sum\limits_{n = 1}^N {(area\,of\,{R_n})} $ < 24, for each positive integer N.

D

The length of latusrectum of E₉ is ${1 \over 6}$

Question 6: Palette 6/32

JEE Advanced 2018 MSQ

Consider two straight lines, each of which is tangent to both the circle x² + y² = (1/2) and the parabola y² = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre 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 1$\sqrt 2 $ and the length of the latus rectum is 1

B

For the ellipse, the eccentricity is 1/2 and the length of the latus rectum is 1/2

C

The area of the region bounded by the ellipse between the lines $x = {1 \over {\sqrt 2 }}$ and x = 1 is ${1 \over {4\sqrt 2 }}(\pi - 2)$

D

The area of the region bounded by the ellipse between the lines $x = {1 \over {\sqrt 2 }}$ and x = 1 is ${1 \over {16}}(\pi - 2)$

Question 7: Palette 7/32

JEE Advanced 2018 MCQ

Let S be the circle in the XY-plane defined the equation x² + y² = 4.

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

A

(x + y)² = 3xy

B

x^2/3 + y^2/3 = 2^4/3

C

x² + y² = 2xy

D

x² + y² = x²y²

Question 8: Palette 8/32

JEE Advanced 2016 MCQ

Let ${F_1}\left( {{x_1},0} \right)$ and ${F_2}\left( {{x_2},0} \right)$ for ${{x_1} < 0}$ and ${{x_2} > 0}$, be the foci of the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$. 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

$\left( { - {9 \over {10}},0} \right)$

B

$\left( { {2 \over {3}},0} \right)$

C

$\left( { {9 \over {10}},0} \right)$

D

$\left( {{2 \over 3},\sqrt 6 } \right)$

Question 9: Palette 9/32

JEE Advanced 2016 MCQ

Let ${F_1}\left( {{x_1},0} \right)$ and ${F_2}\left( {{x_2},0} \right)$ for ${{x_1} < 0}$ and ${{x_2} > 0}$, be the foci of the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$. 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 $M{F_1}N{F_2}$is

A

$3:4$

B

$4:5$

C

$5:8$

D

$2:3$

Question 10: Palette 10/32

JEE Advanced 2015 MSQ

Let ${E_1}$ and ${E_2}$ be two ellipses whose centres 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 ${x^2} + {\left( {y - 1} \right)^2} = 2$. The straight line $x+y=3$ touches the curves $S$, ${E_1}$ and ${E_2}$ at $P, Q$ and $R$ respectively. Suppose that $PQ = PR = {{2\sqrt 2 } \over 3}$. If ${e_1}$ and ${e_2}$ are the eccentricities of ${E_1}$ and ${E_2}$, respectively, then the correct expression(s) is (are)

A

$\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 = {{43} \over {40}}$

B

${e_1}{e_2} = {{\sqrt 7 } \over {2\sqrt {10} }}$

C

$\left| {\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 } \right| = {5 \over 8}$

D

${e_1}{e_2} = {{\sqrt 3 } \over 4}$

Question 11: Palette 11/32

JEE Advanced 2014 MCQ

The common tangents to the circle ${x^2} + {y^2} = 2$ and the parabola ${y^2} = 8x$ touch the circle at the points $P, Q$ and the parabola at the points $R$, $S$. Then the area of the quadrilateral $PQRS$ is

A

$3$

B

$6$

C

$9$

D

$15$

Question 12: Palette 12/32

JEE Advanced 2012 MCQ

The ellipse ${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse ${E_2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse ${E_2}$ is

A

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

B

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

C

${{1 \over 2}}$

D

${{3 \over 4}}$

Question 13: Palette 13/32

JEE Advanced 2010 MCQ

Tangents are drawn from the point $P(3, 4)$ to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ touching the ellipse at points $A$ and $B$.

The coordinates of $A$ and $B$ are

A

$(3,0)$ and $(0,2)$

B

$\left( { - {8 \over 5},{{2\sqrt {161} } \over {15}}} \right)$ and $\left( { - {9 \over 5},{8 \over 5}} \right)$

C

$\left( { - {8 \over 5},{{2\sqrt {161} } \over {15}}} \right)$ and $(0,2)$

D

$(3,0)$ and $\left( { - {9 \over 5},{8 \over 5}} \right)$

Question 14: Palette 14/32

JEE Advanced 2010 MCQ

Tangents are drawn from the point $P(3, 4)$ to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ touching the ellipse at points $A$ and $B$.

The equation of the locus of the point whose distances from the point $P$ and the line $AB$ are equal, is

A

$9{x^2} + {y^2} - 6xy - 54x - 62y + 241 = 0$

B

${x^2} + 9{y^2} + 6xy - 54x + 62y - 241 = 0$

C

$9{x^2} + 9{y^2} - 6xy - 54x - 62y - 241 = 0$

D

${x^2} + {y^2} - 2xy + 27x + 31y - 120 = 0$

Question 15: Palette 15/32

JEE Advanced 2010 MCQ

Tangents are drawn from the point $P(3, 4)$ to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ touching the ellipse at points $A$ and $B$.

The orthocentre of the triangle $PAB$ is

A

$\left( {5,{8 \over 7}} \right)$

B

$\left( {{7 \over 5},{{25} \over 8}} \right)$

C

$\left( {{11 \over 5},{{8} \over 5}} \right)$

D

$\left( {{8 \over 25},{{7} \over 5}} \right)$

Question 16: Palette 16/32

JEE Advanced 2009 MSQ

An ellipse intersects the hyperbola $2{x^2} - 2{y^2} = 1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes then

A

equation of ellipse is ${x^2} + 2{y^2} = 2$

B

the foci of ellipse are $\left( { \pm 1,0} \right)$

C

equation of ellipse is ${x^2} + 2{y^2} = 4$

D

the foci of ellipse are $\left( { \pm \sqrt 2 ,0} \right)$

Question 17: Palette 17/32

JEE Advanced 2009 MSQ

In a triangle $ABC$ with fixed base $BC$, the vertex $A$ moves such that $$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$

If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$, respectively, then

A

$b+c=4a$

B

$b+c=2a$

C

locus of point $A$ is an ellipse

D

locus of point $A$ is a pair of straight lines

Question 18: Palette 18/32

JEE Advanced 2009 MCQ

The normal at a point $P$ on the ellipse ${x^2} + 4{y^2} = 16$ meets the $x$- axis $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points

A

$\left( { \pm {{3\sqrt 5 } \over 2},\, \pm {2 \over 7}} \right)$

B

$\left( { \pm {{3\sqrt 5 } \over 2},\, \pm \sqrt {{{19} \over 4}} } \right)$

C

$\left( { \pm 2\sqrt 3 , \pm {1 \over 7}} \right)$

D

$\left( { \pm 2\sqrt 3 , \pm {{4\sqrt 3 } \over 7}} \right)$

Question 19: Palette 19/32

JEE Advanced 2009 MCQ

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse ${x^2} + 9{y^2} = 9$ meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A$, $M$ and the origin $O$ is

A

${{31} \over {10}}$

B

${{29} \over {10}}$

C

${{21} \over {10}}$

D

${{27} \over {10}}$

Question 20: Palette 20/32

JEE Advanced 2009 MCQ

Match the conics in Column I with the statements/expressions in Column II :

	Column I		Column II
(A)	Circle	(P)	The locus of the point ($h,k$) for which the line $hx+ky=1$ touches the circle $x^2+y^2=4$.
(B)	Parabola	(Q)	Points z in the complex plane satisfying $|z+2|-|z-2|=\pm3$.
(C)	Ellipse	(R)	Points of the conic have parametric representation $x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),y = {{2t} \over {1 + {t^2}}}$
(D)	Hyperbola	(S)	The eccentricity of the conic lies in the interval $1 \le x \le \infty $.
		(T)	Points z in the complex plane satisfying ${\mathop{\rm Re}\nolimits} {(z + 1)^2} = |z{|^2} + 1$.

A

(A)$\to$(P); (B)$\to$(S), (T); (C)$\to$(R); (D)$\to$(R), (S)

B

(A)$\to$(P); (B)$\to$(S), (T); (C)$\to$(R); (D)$\to$(Q), (S)

C

(A)$\to$(P); (B)$\to$(S), (T); (C)$\to$(S); (D)$\to$(R), (S)

D

(A)$\to$(P); (B)$\to$(P), (T); (C)$\to$(R); (D)$\to$(Q), (S)

Question 21: Palette 21/32

JEE Advanced 2008 MSQ

Let $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,$ be the end points of the latus rectum of the ellipse ${x^2} + 4{y^2} = 4.$ The equations of parabolas with latus rectum $PQ$ are :

A

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

B

${x^2} - 2\sqrt 3y = 3 + \sqrt 3 $

C

${x^2} + 2\sqrt 3y = 3 - \sqrt 3 $

D

${x^2} - 2\sqrt 3 y = 3 - \sqrt 3 $

Question 22: Palette 22/32

JEE Advanced 2008 MCQ

Consider the two curves ${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$. Then,

A

${C_1}$ and ${C_2}$ touch each other only at one point.

B

${C_1}$ and ${C_2}$ touch each other exactly at two points

C

${C_1}$ and ${C_2}$ intersect (but do not touch ) at exactly two points

D

${C_1}$ and ${C_2}$ neither intersect nor touch each other

Question 23: Palette 23/32

JEE Advanced 2005 MCQ

The minimum area of triangle formed by the tangent to the ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ and coordinate axes is

A

$ab$ sq. units

B

${{{{a^2} + {b^2}} \over 2}}$ sq. units

C

${{{{\left( {a + b} \right)}^2}} \over 2}$ sq. units

D

${{{a^2} + ab + {b^2}} \over 3}$ sq. units

Question 24: Palette 24/32

JEE Advanced 2005 MCQ

Find the equation of the common tangent in the first quadrant to the circle $x^{2}+y^{2}=16$ and the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$. Also find the length of the intercept of the tangent between the coordinate axes.

A

$\frac{14}{\sqrt5}$

B

$\frac{5}{\sqrt3}$

C

$\frac{14}{\sqrt3}$

D

$\frac{15}{\sqrt3}$

Question 25: Palette 25/32

JEE Advanced 2004 MCQ

If tangents are drawn to the ellipse ${x^2} + 2{y^2} = 2,$ then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

A

${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$

B

${1 \over {4{x^2}}} + {1 \over {2{y^2}}} = 1$

C

${{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$

D

${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$

Question 26: Palette 26/32

JEE Advanced 2003 MCQ

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1,$ is

A

$27/4$ sq. units

B

$9$ sq. units

C

$27/2$ sq. units

D

$27$ sq. units

Question 27: Palette 27/32

JEE Advanced 1999 MSQ

On the ellipse $4{x^2} + 9{y^2} = 1,$ the points at which the tangents are parallel to the line $8x = 9y$ are

A

$\left( {{2 \over 5},{1 \over 5}} \right)$

B

$\left( -{{2 \over 5},{1 \over 5}} \right)$

C

$\left( -{{2 \over 5},-{1 \over 5}} \right)$

D

$\left( {{2 \over 5},-{1 \over 5}} \right)$

Question 28: Palette 28/32

JEE Advanced 1998 MCQ

If $P=(x, y)$, ${F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)$ and $16{x^2} + 25{y^2} = 400,$ then $P{F_1} + P{F_2}$ equals

A

$8$

B

$6$

C

$10$

D

$12$

Question 29: Palette 29/32

JEE Advanced 1998 MCQ

The number of values of $c$ such that the straight line $y=4x + c$ touches the curve $\left( {{x^2}/4} \right) + {y^2} = 1$ is

A

$0$

B

$1$

C

$2$

D

infinite.

Question 30: Palette 30/32

JEE Advanced 1995 MCQ

The radius of the circle passing through the foci of the ellipse ${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$, and having its centre at $(0, 3)$ is

A

$4$

B

$3$

C

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

D

${{7 \over 2}}$

Question 31: Palette 31/32

JEE Advanced 1994 MCQ

The equation $2{x^2} + 3{y^2} - 8x - 18y + 35 = k$ represents

A

no locus if $k > 0$

B

an ellipse if $k < 0$

C

a point if $k = 0$

D

a hyperbola if $k > 0$

Question 32: Palette 32/32

JEE Advanced 1994 MCQ

Let $E$ be the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ and $C$ be the circle ${x^2} + {y^2} = 9$. Let $P$ and $Q$ be the points $(1, 2)$ and $(2, 1)$ respectively. Then

A

$Q$ lies inside $C$ but outside $E$

B

$Q$ lies outside both $C$ and $E$

C

$P$ lies inside both $C$ and $E$

D

$P$ lies inside $C$ but outside $E$