Question 1: Palette 1/47

JEE Advanced 2026 MCQ

Let T be the tangent to the parabola $y^2 = 16x$ at the point $(64, 32)$. Let L be the tangent to the same parabola at another point $(x_1, y_1)$ on the parabola. If L and T are perpendicular to each other, then the distance between the point $(x_1, y_1)$ and the focus of the parabola, is :

A

$ \frac{15}{4} $

B

4

C

$ \frac{17}{4} $

D

5

Question 2: Palette 2/47

JEE Advanced 2025 MSQ

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$, such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $\mathcal{R}$ denote the region lying in the first quadrant, enclosed by the parabola $y^2=x$, the curve $S$, and the lines $x=1$ and $x=4$.

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

A

$(4, \sqrt{3}) \in S$

B

$(5, \sqrt{2}) \in S$

C

Area of $\mathcal{R}$ is $\frac{14}{3} - 2\sqrt{3}$

D

Area of $\mathcal{R}$ is $\frac{14}{3} - \sqrt{3}$

Question 3: Palette 3/47

JEE Advanced 2024 MSQ

Let $A_1, B_1, C_1$ be three points in the $x y$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8 x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statements is (are) TRUE?

A

The length of the line segment $O A_1$ is $4 \sqrt{3}$

B

The length of the line segment $A_1 B_1$ is 16

C

The orthocenter of the triangle $A_1 B_1 C_1$ is $(0,0)$

D

The orthocenter of the triangle $A_1 B_1 C_1$ is $(1,0)$

Question 4: Palette 4/47

JEE Advanced 2023 MCQ

Let $P$ be a point on the parabola $y^2=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is

A

$(2,3)$

B

$(1,3)$

C

$(2,4)$

D

$(3,4)$

Question 5: Palette 5/47

JEE Advanced 2022 MSQ

Consider the parabola $y^{2}=4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_{1}$ and $P_{2}$. Let $Q_{1}$ and $Q_{2}$ be points on the lines $S P_{1}$ and $S P_{2}$ respectively such that $P Q_{1}$ is perpendicular to $S P_{1}$ and $P Q_{2}$ is perpendicular to $S P_{2}$. Then, which of the following is/are TRUE?

A

$S Q_{1}=2$

B

$Q_{1} Q_{2}=\frac{3 \sqrt{10}}{5}$

C

$P Q_{1}=3$

D

$S Q_{2}=1$

Question 6: Palette 6/47

JEE Advanced 2021 MSQ

Let E denote the parabola y² = 8x. Let P = ($-$2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?

A

The triangle PFQ is a right-angled triangle

B

The triangle QPQ' is a right-angled triangle

C

The distance between P and F is 5$\sqrt 2 $

D

F lies on the line joining Q and Q'

Question 7: Palette 7/47

JEE Advanced 2020 MCQ

Let a, b and $\lambda $ be positive real numbers. Suppose P is an end point of the latus return of the
parabola y² = 4$\lambda $x, and suppose the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ 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

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

B

${{1 \over 2}}$

C

${{1 \over 3}}$

D

${{2 \over 5}}$

Question 8: Palette 8/47

JEE Advanced 2019 MCQ

Let the circles

C₁ : x² + y² = 9 and C₂ : (x $-$ 3)² + (y $-$ 4)² = 16, intersect at the points X and Y. Suppose that another circle C₃ : (x $-$ h)² + (y $-$ k)² = r² satisfies the following conditions :

(i) Centre of C₃ is collinear with the centres of C₁ and C₂.

(ii) C₁ and C₂ both lie inside C₃ and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8$\alpha $y.

There are some expression given in the List-I whose values are given in List-II below.



Which of the following is the only INCORRECT combination?

A

(III), (R)

B

(IV), (S)

C

(I), (P)

D

(IV), (U)

Question 9: Palette 9/47

JEE Advanced 2019 MCQ

Let the circle C₁ : x² + y² = 9 and C₂ : (x $-$ 3)² + (y $-$ 4)² = 16, intersect at the points X and Y. Suppose that another circle C₃ : (x $-$ h)² + (y $-$ k)² = r² satisfies the following conditions :

(i) centre of C₃ is collinear with the centers of C₁ and C₂.

(ii) C₁ and C₂ both lie inside C₃, and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8$\alpha $y.

There are some expression given in the List-I whose values are given in List-II below.



Which of the following is the only CORRECT combination?

A

(II), (T)

B

(I), (S)

C

(II), (Q)

D

(I), (U)

Question 10: Palette 10/47

JEE Advanced 2017 MCQ

If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?

A

(III) (i) (P)

B

(I) (ii) (Q)

C

(II) (iv) (R)

D

(III) (ii) (Q)

Question 11: Palette 11/47

JEE Advanced 2017 MCQ

If a chord, which is not a tangent, of the parabola y² = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) possible value(s) of p, h and k?

A

p = $-$1, h = 1, k = $-$3

B

p = 2, h = 3, k = $-$4

C

p = $-$2, h = 2, k = $-$4

D

p = 5, h = 4, k = $-$3

Question 12: Palette 12/47

JEE Advanced 2016 MSQ

Let $P$ be the point on the parabola ${y^2} = 4x$ which is at the shortest distance from the center $S$ of the circle ${x^2} + {y^2} - 4x - 16y + 64 = 0$. Let $Q$ be the point on the circle dividing the line segment $SP$ internally. Then

A

$SP = 2\sqrt 5 $

B

$SQ:QP = \left( {\sqrt 5 + 1} \right):2$

C

the $x$-intercept of the normal to the parabola at $P$ is $6$

D

the slope of the tangent to the circle at $Q$ is ${1 \over 2}$

Question 13: Palette 13/47

JEE Advanced 2016 MSQ

The circle ${C_1}:{x^2} + {y^2} = 3,$ with centre at $O$, intersects the parabola ${x^2} = 2y$ at the point $P$ in the first quadrant, Let the tangent to the circle ${C_1}$, at $P$ touches other two circles ${C_2}$ and ${C_3}$ at ${R_2}$ and ${R_3}$, respectively. Suppose ${C_2}$ and ${C_3}$ have equal radil ${2\sqrt 3 }$ and centres ${Q_2}$ and ${Q_3}$, respectively. If ${Q_2}$ and ${Q_3}$ lie on the $y$-axis, then

A

${Q_2}{Q_3} = 12$

B

${R_2}{R_3} = 4\sqrt 6 $

C

area of the triangle $O{R_2}{R_3}$ is $6\sqrt 2 $

D

area of the triangle $P{Q_2}{Q_3}$ is $4\sqrt 2 $

Question 14: Palette 14/47

JEE Advanced 2015 MSQ

Let $P$ and $Q$ be distinct points on the parabola ${y^2} = 2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta OPQ$ is ${3\sqrt 2 ,}$ then which of the following is (are) the coordinates of $P$?

A

$\left( {4,2\sqrt 2 } \right)$

B

$\left( {9,3\sqrt 2 } \right)$

C

$\left( {{1 \over 4},{1 \over {\sqrt 2 }}} \right)$

D

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

Question 15: Palette 15/47

JEE Advanced 2014 MCQ

Let $a, r, s, t$ be nonzero real numbers. Let $P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$ and $S\,\,\left( {a{s^2},2as} \right)$ be distinct points on the parabola ${y^2} = 4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a,0)$

If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

A

${{{{\left( {{t^2} + 1} \right)}^2}} \over {2{t^3}}}$

B

${{a{{\left( {{t^2} + 1} \right)}^2}} \over {2{t^3}}}$

C

${{a{{\left( {{t^2} + 1} \right)}^2}} \over {{t^3}}}$

D

${{a{{\left( {{t^2} + 2} \right)}^2}} \over {{t^3}}}$

Question 16: Palette 16/47

JEE Advanced 2014 MCQ

Let $a, r, s, t$ be nonzero real numbers. Let $P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$ and $S\,\,\left( {a{s^2},2as} \right)$ be distinct points on the parabola ${y^2} = 4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a,0)$

The value of $r$ is

A

$ - {1 \over t}$

B

${{{t^2} + 1} \over t}$

C

$ {1 \over t}$

D

${{{t^2} - 1} \over t}$

Question 17: Palette 17/47

JEE Advanced 2013 MCQ

A line $L:y=mx+3$ meets $y$-axis at R$(0, 3)$ and the arc of the parabola ${y^2} = 16x,$ $0 \le y \le 6$ at the point $F\left( {{x_0},{y_0}} \right)$. The tangent to the parabola at $F\left( {{x_0},{y_0}} \right)$ intersects the $y$-axis at $G\left( {0,{y_1}} \right)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $EFG$ has a local maximum.

Match List $I$ with List $II$ and select the correct answer using the code given below the lists:

List $I$
P.$\,\,\,m = $
Q.$\,\,\,$Maximum area of $\Delta EFG$ is
R.$\,\,\,$ ${y_0} = $
S.$\,\,\,$ ${y_1} = $

List $II$
1.$\,\,\,$ ${1 \over 2}$
2.$\,\,\,$ $4$
3.$\,\,\,$ $2$
4.$\,\,\,$ $1$

A

$P = 4,Q = 1,R = 2,S = 3$

B

$P = 3,Q = 4,R = 1,S = 2$

C

$P = 1,Q = 3,R = 2,S = 4$

D

$P = 1,Q = 3,R = 4,S = 2$

Question 18: Palette 18/47

JEE Advanced 2013 MCQ

Let $PQ$ be a focal chord of the parabola ${y^2} = 4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a$, $a>0$.

Length of chord $PQ$ is

A

$7a$

B

$5a$

C

$2a$

D

$3a$

Question 19: Palette 19/47

JEE Advanced 2013 MCQ

Let $PQ$ be a focal chord of the parabola ${y^2} = 4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a$, $a>0$.

If chord $PQ$ subtends an angle $\theta $ at the vertex of ${y^2} = 4ax$, then tan $\theta = $

A

${2 \over 3}\sqrt 7 $

B

${-2 \over 3}\sqrt 7 $

C

${2 \over 3}\sqrt 5 $

D

${-2 \over 3}\sqrt 5 $

Question 20: Palette 20/47

JEE Advanced 2011 MCQ

Let $(x, y)$ be any point on the parabola ${y^2} = 4x$. Let $P$ be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1 : 3$. Then the locus of $P$ is

A

${x^2} = y$

B

${y^2} = 2x$

C

${y^2} = x$

D

${x^2} = 2y$

Question 21: Palette 21/47

JEE Advanced 2011 MSQ

Let L be a normal to the parabola y² = 4x. If L passes through the point (9, 6), then L is given by

A

y $-$ x + 3 = 0

B

y + 3x $-$ 33 = 0

C

y + x $-$ 15 = 0

D

7 $-$ 2x + 12 = 0

Question 22: Palette 22/47

JEE Advanced 2010 MSQ

Let $A$ and $B$ be two distinct points on the parabola ${y^2} = 4x$. If the axis of the parabola touches a circle of radius $r$ having $AB$ as its diameter, then the slope of the line joining $A$ and $B$ can be

A

$ - {1 \over r}$

B

$ {1 \over r}$

C

$ {2 \over r}$

D

$ - {2 \over r}$

Question 23: Palette 23/47

JEE Advanced 2009 MCQ

The locus of the orthocentre of the triangle formed by the lines

$(1 + p)x - py + p(1 + p) = 0, $

$(1 + q)x - qy + q(1 + q) = 0$

and $y = 0$, where $p \ne q$, is :

A

a hyperbola.

B

a parabola.

C

an ellipse.

D

a straight line.

Question 24: Palette 24/47

JEE Advanced 2009 MSQ

The tangent $PT$ and the normal $PN$ to the parabola ${y^2} = 4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose

A

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

B

directrix is $x=0$

C

latus rectum is ${{{2a} \over 3}}$

D

focus is $(a, 0)$

Question 25: Palette 25/47

JEE Advanced 2007 MCQ

STATEMENT-1: The curve $y = {{ - {x^2}} \over 2} + x + 1$ is symmetric with respect to the line $x=1$. because

STATEMENT-2: A parabola is symmetric about its axis.

A

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is True.

Question 26: Palette 26/47

JEE Advanced 2007 MCQ

Consider the circle ${x^2} + {y^2} = 9$ and the parabola ${y^2} = 8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangent to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x$-axis at $S$.

The ratio of the areas of the triangles $PQS$ and $PQR$ is

A

$1:\sqrt 2 $

B

$1:2$

C

$1:4$

D

$1:8$

Question 27: Palette 27/47

JEE Advanced 2007 MCQ

Consider the circle ${x^2} + {y^2} = 9$ and the parabola ${y^2} = 8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangent to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x$-axis at $S$.

The radius of the circumcircle of the triangle $PRS$ is

A

$5$

B

$3\sqrt 3 $

C

$3\sqrt 2 $

D

$2\sqrt 3 $

Question 28: Palette 28/47

JEE Advanced 2007 MCQ

Consider the circle ${x^2} + {y^2} = 9$ and the parabola ${y^2} = 8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangent to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x$-axis at $S$.

The radius of the incircle of the triangle $PQR$ is

A

$4$

B

$3$

C

${8 \over 3}$

D

$2$

Question 29: Palette 29/47

JEE Advanced 2007 MCQ

STATEMENT - 1 : The curve $y=\frac{-x^{2}}{2}+x+1$ is symmetric with respect to the line $x=1$.

STATEMENT - 2 : A parabola is symmetric about its axis.

A

Statement- 1 is True, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is true; Statement- 2 is NOT a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is True

Question 30: Palette 30/47

JEE Advanced 2007 MCQ

The tangent to the curve $y=e^x$ drawn at the point ($c,e^c$) intersects the line joining the points ($c-1,e^{c-1}$) and ($c+1,e^{c+1}$)

A

on the left of $x=c$

B

on the right of $x=c$

C

at no point

D

at all points

Question 31: Palette 31/47

JEE Advanced 2007 MCQ

The ratio of the areas of the triangles PQS and PQR is

A

1 : $\sqrt2$

B

1 : 2

C

1 : 4

D

1 : 8

Question 32: Palette 32/47

JEE Advanced 2007 MCQ

The radius of the circumcircle of the triangle PRS is

A

5 units

B

3$\sqrt3$ units

C

3$\sqrt2$ units

D

2$\sqrt3$ units

Question 33: Palette 33/47

JEE Advanced 2007 MCQ

The radius of the incircle of the triangle PQR is

A

4 units

B

3 units

C

$\frac{8}{3}$ units

D

2 units

Question 34: Palette 34/47

JEE Advanced 2006 MCQ

The axis of a parabola is along the line $y = x$ and the distances of its vertex and focus from origin are $\sqrt 2 $ and $2\sqrt 2 $ respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

A

${\left( {x + y} \right)^2} = \left( {x - y - 2} \right)$

B

${\left( {x - y} \right)^2} = \left( {x + y - 2} \right)$

C

${\left( {x - y} \right)^2} = 4\left( {x + y - 2} \right)$

D

${\left( {x - y} \right)^2} = 8\left( {x + y - 2} \right)$

Question 35: Palette 35/47

JEE Advanced 2006 MCQ

$ \text { Normals are drawn at points } \mathrm{P}, \mathrm{Q} \text { and } \mathrm{R} \text { lying on the parabola } y^2=4 x \text { which intersect at }(3,0) \text {. Then } $

(i)	Area of $\triangle \mathrm{PQR}$	(A)	2
(ii)	Radius of circumcircle of $\triangle \mathrm{PQR}$	(B)	5/2
(iii)	Centroid of $\triangle \mathrm{PQR}$	(C)	(5/2,0)
(iv)	Circumcentre of $\triangle \mathrm{PQR}$	(D)	(2/3,0)

A

$ \begin{aligned} & \text { (i)-(A); (ii)-(B); (iii)-(D); } \text { (iv)-(C) } \end{aligned} $

B

$ \begin{aligned} & \text { (i)-(B); (ii)-(A); (iii)-(D); } \text { (iv)-(C) } \end{aligned} $

C

$ \begin{aligned} & \text { (i)-(A); (ii)-(B); (iii)-(C); } \text { (iv)-(D) } \end{aligned} $

D

$ \begin{aligned} & \text { (i)-(A); (ii)-(D); (iii)-(B); } \text { (iv)-(C) } \end{aligned} $

Question 36: Palette 36/47

JEE Advanced 2006 MSQ

The equations of the common tangents to the parabola $y = {x^2}$ and $y = - {\left( {x - 2} \right)^2}$ is/are

A

$y = 4\left( {x - 1} \right)$

B

$y=0$

C

$y = - 4\left( {x - 1} \right)$

D

$y = - 30x - 50$

Question 37: Palette 37/47

JEE Advanced 2005 MCQ

Tangent to the curve $y = {x^2} + 6$ at a point $(1, 7)$ touches the circle ${x^2} + {y^2} + 16x + 12y + c = 0$ at a point $Q$. Then the coordinates of $Q$ are

A

$(-6, -11)$

B

$(-9, -13)$

C

$(-10, -15)$

D

$(-6, -7)$

Question 38: Palette 38/47

JEE Advanced 2004 MCQ

The angle between the tangents drawn from the point $(1, 4)$ to the parabola ${y^2} = 4x$ is

A

$\pi /6$

B

$\pi /4$

C

$\pi /3$

D

$\pi /2$

Question 39: Palette 39/47

JEE Advanced 2003 MCQ

The focal chord to ${y^2} = 16x$ is tangent to ${\left( {x - 6} \right)^2} + {y^2} = 2,$ then the possible values of the slope of the chord, are

A

$\left\{ { - 1,\,1} \right\}$

B

$\left\{ { - 2,\,2} \right\}$

C

$\left\{ { - 2,\,-1/2} \right\}$

D

$\left\{ { 2,\,-1/2} \right\}$

Question 40: Palette 40/47

JEE Advanced 2002 MCQ

The equation of the common tangent to the curves ${y^2} = 8x$ and $xy = - 1$ is

A

$3y = 9x + 2$

B

$y = 2x + 1$

C

$2y = x + 8$

D

$y= x + 2$

Question 41: Palette 41/47

JEE Advanced 2002 MCQ

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola ${y^2} = 4ax$ is another parabola with directrix

A

$x = -a$

B

$x = -a/2$

C

$x = 0$

D

$x = a/2$

Question 42: Palette 42/47

JEE Advanced 2001 MCQ

The equation of the common tangent touching the circle ${\left( {x - 3} \right)^2} + {y^2} = 9$ and the parabola ${y^2} = 4x$ above the $x$-axis is

A

$\sqrt {3y} = 3x + 1$

B

$\sqrt {3y} = - \left( {x + 3} \right)$

C

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

D

$\sqrt {3y} = - \left( {3x + 1} \right)$

Question 43: Palette 43/47

JEE Advanced 2001 MCQ

The equation of the directrix of the parabola ${y^2} + 4y + 4x + 2 = 0$

A

$x = - 1$

B

$x = 1$

C

$x = - 3/2$

D

$x = 3/2$

Question 44: Palette 44/47

JEE Advanced 2000 MCQ

If the line $x - 1 = 0$ is the directrix of the parabola ${y^2} - kx + 8 = 0,$ then one of the values of $k$ is

A

$1/8$

B

$8$

C

$4$

D

$1/4$

Question 45: Palette 45/47

JEE Advanced 2000 MCQ

If $x + y = k$ is normal to ${y^2} = 12x,$ then $k$ is

A

$3$

B

$9$

C

$-9$

D

$-3$

Question 46: Palette 46/47

JEE Advanced 1999 MCQ

The curve described parametrically by $x = {t^2} + t + 1,$ $y = {t^2} - t + 1 $ represents

A

a pair of straight lines

B

an ellipse

C

a parabola

D

a hyperbola

Question 47: Palette 47/47

JEE Advanced 1995 MCQ

Consider a circle with its centre lying on the focus of the parabola ${y^2} = 2px$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

A

$\left( {{p \over 2},p} \right)$ or $\left( {{p \over 2},- p} \right)$

B

$\left( { {p \over 2}, {p \over 2}} \right)$

C

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

D

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