How many questions from Parabola appear in JEE?

Parabola has appeared in 101 of the last 186 JEE Main and JEE Advanced papers — about 54% of them — contributing 121 questions in total across those papers.

Is Parabola an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 54% 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 Parabola 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.

Parabola JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Advanced 2026

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) $\dfrac{15}{4}$
(B) $4$
(C) $\dfrac{17}{4}$
(D) $5$

Q2·MathematicsSingle correctJEE Main 2026

Let O be the vertex of the parabola

y^2 = 4x

and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:

(A)1
(B)2
(C)4
(D)8

Q3·MathematicsSingle correctJEE Main 2026

Let chord PQ of length 3√13 of the parabola y

^2

= 12x be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle α at the focus of the parabola, then sin α is equal to:

(A) $\frac{3}{5}$
(B) $\frac{4}{5}$
(C) $\frac{5}{13}$
(D) $\frac{12}{13}$

Q4·MathematicsSingle correctJEE Main 2026

Let the directrix of the parabola

P : y^2 = 8x

, cut x-axis at the point A. Let

B(α, β)

,

α > 1

, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of

\triangle ABC

is :

(A)80
(B)160
(C)174
(D)192

Q5·MathematicsNumericalJEE Main 2026

Let

A, B

and

C

be the vertices of a variable right angled triangle inscribed in the parabola

y^{2} = 16x

. Let the vertex

B

containing the right angle be

(4, 8)

and the locus of the centroid of

\triangle ABC

be a conic

C_{o}

. Then three times the length of latus rectum of

C_{o}

is ______

Q6·MathematicsSingle correctJEE Main 2026

Let the parabola

y = x^2 + px + q

passing through the point

(1, -1)

be such that the distance between its vertex and the

x

-axis is minimum. Then the value of

p^2 + q^2

is:

(A) $2$
(B) $4$
(C) $5$
(D) $8$

Q7·MathematicsSingle correctJEE Main 2026

Let A be the focus of the parabola

y^{2} = 8x

. Let the line

y = mx + c

intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is

\left(\dfrac{7}{3}, \dfrac{4}{3}\right)

, then

(BC)^{2}

is equal to :

(A)41
(B)80
(C)89
(D)32

Q8·MathematicsSingle correctJEE Main 2026

Let the image of parabola

x^{2} = 4y

, in the line

x - y = 1

be

(y + \alpha)^{2} = b(x - c)

, a, b, c ∈ ℕ. Then

a + b + c

is equal to

(A)12
(B)4
(C)6
(D)8

Q9·MathematicsSingle correctJEE Main 2026

An equilateral triangle OAB is inscribed in the parabola

y^2 = 4x

with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

(A) $4(3 - \sqrt{3})$
(B) $2(8 - 3\sqrt{3})$
(C) $4(6 + \sqrt{3})$
(D) $2(3 + \sqrt{3})$

Q10·MathematicsSingle correctJEE Main 2026

If the chord joining the points

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

and

P_{2}(x_{2}, y_{2})

on the parabola

y^{2} = 12x

subtends a right angle at the vertex of the parabola, then

x_{1}x_{2} - y_{1}y_{2}

is equal to

(A)288
(B)280
(C)284
(D)292

Q11·MathematicsSingle correctJEE Main 2026

Let the locus of the mid-point of the chord through the origin O of the parabola

y^2 = 4x

be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :

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

Q12·MathematicsSingle correctJEE Main 2026

Let one end of a focal chord of the parabola

y^2 = 16x

be (16, 16). If P(α, β) divides this focal chord internally in the ratio 5 : 2, then the minimum value of α + β is equal to :

(A)22
(B)7
(C)5
(D)16

Q13·MathematicsSingle correctJEE Main 2026

Let

y^2 = 12x

be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that ∠OPA = 90°. Then the locus of the centroid of such triangles OPA is :

(A) $y^2 - 6x + 4 = 0$
(B) $y^2 - 9x + 6 = 0$
(C) $y^2 - 2x + 8 = 0$
(D) $y^2 - 4x + 8 = 0$

Q14·MathematicsSingle correctJEE Main 2025

Let P be the parabola, whose focus is

(-2,1)

and directrix is

2x+y+2=0

. Then the sum of the ordinates of the points on P, whose abscissa is

-2

, is

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

Q15·MathematicsSingle correctJEE Main 2025

The axis of a parabola is the line

y=x

and its vertex and focus are in the first quadrant at distances

\sqrt2

and

2\sqrt2

units from the origin, respectively. If the point

(1,k)

lies on the parabola, then a possible value of

k

is:

(A)4
(B)9
(C)3
(D)8

Q16·MathematicsSingle correctJEE Main 2025

A line passing through the point

A(-2,0)

, touches the parabola

P:y^2=x-2

at the point

B

in the first quadrant. The area of the region bounded by the line

AB

, parabola

P

and the x-axis, is:

(A) $\dfrac{7}{3}$
(B)2
(C) $\dfrac{8}{3}$
(D)3

Q17·MathematicsSingle correctJEE Main 2025

The shortest distance between the curves

y^2=8x

and

x^2+y^2+12y+35=0

is:

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

Q18·MathematicsSingle correctJEE Main 2025

Let the focal chord

PQ

of the parabola

y^2=4x

make an angle of

60^\circ

with the positive x-axis, where

P

lies in the first quadrant. If the circle, whose one diameter is

PS

,

S

being the focus of the parabola, touches the y-axis at the point

(0,\alpha)

, then

5\alpha^2

is equal to:

(A)15
(B)25
(C)30
(D)20

Q19·MathematicsSingle correctJEE Main 2025

Let one focus of the hyperbola

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

be at

(\sqrt{10},0)

and the corresponding directrix be

x=\dfrac{9}{\sqrt{10}}

. If

e

and

l

respectively are the eccentricity and the length of the latus rectum of

H

, then

9(e^2+l)

is equal to:

(A)14
(B)15
(C)16
(D)12

Q20·MathematicsSingle correctJEE Main 2025

Let the point

P

of the focal chord

PQ

of the parabola

y^2=16x

be

(1,-4)

. If the focus of the parabola divides the chord

PQ

in the ratio

m:n

,

\gcd(m,n)=1

, then

m^2+n^2

is equal to:

(A)17
(B)10
(C)37
(D)26

Q21·MathematicsIntegerJEE Main 2025

Let

y^2=12x

be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that

(SP)(SQ)=\dfrac{147}{4}

. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is

64x^2+64y^2-\alpha x-64\sqrt{3}\,y=\beta

, then

\beta-\alpha

is equal to ______.

Q22·MathematicsSingle correctJEE Main 2025

Two parabolas have the same focus (4,3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersects at the points A and B, then

(AB)^2

is equal to

(A)192
(B)384
(C)96
(D)392

Q23·MathematicsIntegerJEE Main 2025

Let A and B be the two points of intersection of the line

y+5=0

and the mirror image of the parabola

y^2=4x

with respect to the line

x+y+4=0

. If d denotes the distance between A and B, and a denotes the area of

\triangle SAB

, where S is the focus of the parabola

y^2=4x

, then the value of

(a+d)

is ______.

Q24·MathematicsSingle correctJEE Main 2025

If the equation of the parabola with vertex

V\left(\dfrac{3}{2},3\right)

and the directrix

x+2y=0

is

\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0

, then

\alpha+\beta+\gamma

is equal to:

(A)6
(B)8
(C)7
(D)9

Q25·MathematicsSingle correctJEE Main 2025

If the line

3x-2y+12=0

intersects the parabola

4y=3x^2

at the points

A

and

B

, then at the vertex of the parabola, the line segment

AB

subtends an angle equal to

(A) $\tan^{-1}\left(\dfrac{11}{9}\right)$
(B) $\dfrac{\pi}{2}-\tan^{-1}\left(\dfrac{3}{2}\right)$
(C) $\tan^{-1}\left(\dfrac{4}{5}\right)$
(D) $\tan^{-1}\left(\dfrac{9}{7}\right)$

Q26·MathematicsSingle correctJEE Main 2025

Let

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

,

a>b

and

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

. Let the distance between the foci of

E

and the foci of

H

be

2\sqrt{3}

. If

a-A=2

, and the ratio of the eccentricities of

E

and

H

is

\dfrac{1}{3}

, then the sum of the lengths of their latus rectums is equal to:

(A)10
(B)7
(C)8
(D)9

Q27·MathematicsSingle correctJEE Main 2025

Let

P\left(4,4\sqrt{3}\right)

be a point on the parabola

y^2=4ax

and

PQ

be a focal chord of the parabola. If

M

and

N

are the foot of perpendiculars drawn from

P

and

Q

respectively on the directrix of the parabola, then the area of the quadrilateral

PQMN

is equal to:

(A) $\dfrac{263\sqrt{3}}{8}$
(B) $17\sqrt{3}$
(C) $\dfrac{343\sqrt{3}}{8}$
(D) $\dfrac{34\sqrt{3}}{3}$

Q28·MathematicsMultiple correctJEE Advanced 2024

Let

A_1

,

B_1

,

C_1

be three points in the xy-plane. Suppose that the lines

A_1C_1

and

B_1C_1

are tangents to the curve

y^2 = 8x

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 $OA_1$ is $4\sqrt{3}$
(B)The length of the line segment $A_1B_1$ is 16
(C)The orthocenter of the triangle $A_1B_1C_1$ is (0, 0)
(D)The orthocenter of the triangle $A_1B_1C_1$ is (1, 0)

Q29·MathematicsIntegerJEE Advanced 2024

A normal with slope

\frac{1}{\sqrt{6}}

is drawn from the point

(0, -\alpha)

to the parabola

x^2 = -4ay

, where

a > 0

. Let L be the line passing through

(0, -\alpha)

and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B. Let r denote the length of the latus rectum and s denote the square of the length of the line segment AB. If

r : s = 1 : 16

, then the value of

24a

is ______

Q30·MathematicsNumericalJEE Main 2024

Let A, B and C be three points on the parabola

y^{2}=6x

and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. Then

\left(\dfrac{AM\cdot BN}{CD}\right)^{2}

is equal to _______.

Q31·MathematicsNumericalJEE Main 2024

Consider the circle

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

and the parabola

P:y^{2}=8x

. If the set of all values of

\alpha

, for which three chords of the circle C on three distinct lines passing through the point

(\alpha, 0)

are bisected by the parabola P is the interval

(p, q)

, then

(2q-p)^{2}

is equal to _______.

Q32·MathematicsNumericalJEE Main 2024

Let

L_{1},L_{2}

be the lines passing through the point

P(0,1)

and touching the parabola

9x^{2}+12x+18y-14=0

. Let

Q

and

R

be the points on the lines

L_{1}

and

L_{2}

such that

\triangle PQR

is an isosceles triangle with base

QR

. If the slopes of the lines

QR

are

m_{1}

and

m_{2}

, then

16(m_{1}^{2}+m_{2}^{2})

is equal to _______.

Q33·MathematicsNumericalJEE Main 2024

Let a conic

C

pass through the point

(4,-2)

and

P(x,y)

,

x\geq 3

, be any point on

C

. Let the slope of the line touching the conic

C

only at a single point

P

be half the slope of the line joining the points

P

and

(3,-5)

. If the focal distance of the point

(7,1)

on

C

is

d

, then

12d

equals _______.

Q34·MathematicsNumericalJEE Main 2024

Suppose

AB

is a focal chord of the parabola

y^2 = 12x

of length

l

and slope

m < \sqrt{3}

. If the distance of the chord

AB

from the origin is

d

, then

ld^2

is equal to ___.

Q35·MathematicsNumericalJEE Main 2024

Let a line perpendicular to the line

2x-y=10

touch the parabola

y^2=4(x-9)

at the point

P

. The distance of the point

P

from the centre of the circle

x^2+y^2-14x-8y+56=0

is __________.

Q36·MathematicsSingle correctJEE Main 2024

Let PQ be a chord of the parabola

y^2=12x

and the midpoint of PQ be at

(4,1)

. Then, which of the following point lies on the line passing through the points P and Q?

(A) $(3,-3)$
(B) $\left(\tfrac{3}{2},-16\right)$
(C) $(2,-9)$
(D) $\left(\tfrac{1}{2},-20\right)$

Q37·MathematicsNumericalJEE Main 2024

Let the length of the focal chord

PQ

of the parabola

y^2=12x

be

15

units. If the distance of

PQ

from the origin is

p

, then

10p^2

is equal to ___

Q38·MathematicsSingle correctJEE Main 2024

Let P be a parabola with vertex

(2,3)

and directrix

2x+y=6

. Let an ellipse

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

,

a>b

, of eccentricity

\dfrac{1}{\sqrt{2}}

pass through the focus of the parabola P. Then the square of the length of the latus rectum of E, is

(A) $\dfrac{385}{8}$
(B) $\dfrac{347}{8}$
(C) $\dfrac{512}{25}$
(D) $\dfrac{656}{25}$

Q39·MathematicsNumericalJEE Main 2024

Let

P(\alpha,\beta)

be a point on the parabola

y^2=4x

. If

P

also lies on the chord of the parabola

x^2=8y

whose mid point is

\left(1,\dfrac{5}{4}\right)

. Then

(\alpha-28)(\beta-8)

is equal to ___.

Q40·MathematicsSingle correctJEE Advanced 2023

Let P be a point on the parabola

y^2 = 4ax

, where

a > 0

. The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ, 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)

Q41·MathematicsSingle correctJEE Main 2023

Let

PQ

be a focal chord of the parabola

y^{2}=36x

of length

100

, making an acute angle with the positive

x

-axis. Let the ordinate of

P

be positive and

M

be the point on the line segment

PQ

such that

PM:MQ=3:1

. Then which of the following points does NOT lie on the line passing through

M

and perpendicular to the line

PQ

?

(A) $(-3,43)$
(B) $(-6,45)$
(C) $(3,33)$
(D) $(6,29)$

Q42·MathematicsNumericalJEE Main 2023

Let a common tangent to the curves

y^2=4x

and

(x-4)^2+y^2=16

touch the curves at the points P and Q. Then

(PQ)^2

is equal to _________.

Q43·MathematicsSingle correctJEE Main 2023

Let

R

be the focus of the parabola

y^{2}=20x

and the line

y=mx+c

intersect the parabola at two points

P

and

Q

. Let the point

G(10,10)

be the centroid of the triangle

PQR

. If

c-m=6

, then

(PQ)^{2}

is

(A)325
(B)317
(C)296
(D)346

Q44·MathematicsSingle correctJEE Main 2023

Let

A(0,1)

,

B(1,1)

and

C(1,0)

be the mid-points of the sides of a triangle with incentre at the point D. If the focus of the parabola

y^{2}=4ax

passing through D is

(\alpha+\beta\sqrt{2},0)

, where

\alpha

and

\beta

are rational numbers, then

\dfrac{\alpha}{\beta^{2}}

is equal to

(A)6
(B)8
(C)12
(D) $\dfrac{9}{2}$

Q45·MathematicsNumericalJEE Main 2023

The ordinates of the points P and Q on the parabola with focus

(3,0)

and directrix

x=-3

are in the ratio

3:1

. If

R(\alpha,\beta)

is the point of intersection of the tangents to the parabola at P and Q, then

\dfrac{\beta^{2}}{\alpha}

is equal to

Q46·MathematicsNumericalJEE Main 2023

Let the tangent to the curve

x^{2}+2x-4y+9=0

at the point

P(1,3)

on it meet the

y

-axis at

A

. Let the line passing through

P

and parallel to the line

x-3y=6

meet the parabola

y^{2}=4x

at

B

. If

B

lies on the line

2x-3y=8

, then

(AB)^{2}

is equal to _____.

Q47·MathematicsNumericalJEE Main 2023

If the x-intercept of a focal chord of the parabola

y^2=8x+4y+4

is 3, then the length of this chord is equal to

Q48·MathematicsNumericalJEE Main 2023

Let

S

be the set of all

a\in\mathbb{N}

such that the area of the triangle formed by the tangent at the point

P(b,c),\ b,c\in\mathbb{N}

, on the parabola

y^2=2ax

and the lines

x=b,\ y=0

is

16

unit

^2

, then

\displaystyle\sum_{a\in S}a

is equal to

Q49·MathematicsSingle correctJEE Main 2023

The parabolas

ax^2 + 2bx + cy=0

and

dx^2 + 2ex + fy=0

intersect on the line

y=1

. If

a, b, c, d, e, f

are positive real numbers and

a, b, c

are in G.P., then:

(A) $d, e, f$ are in G.P.
(B) $\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c}$ are in A.P.
(C) $d, e, f$ are in A.P.
(D) $\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c}$ are in G.P.

Q50·MathematicsSingle correctJEE Main 2023

Let

A

be a point on the x-axis. Common tangents are drawn from

A

to the curves

x^2 + y^2=8

and

y^2=16x

. If one of these tangents touches the two curves at

Q

and

R

, then

(QR)^2

is equal to:

(A) $81$
(B) $72$
(C) $76$
(D) $64$

Q51·MathematicsSingle correctJEE Main 2023

If

P(h,k)

be a point on the parabola

x=4y^2

, which is nearest to the point

Q(0,33)

, then the distance of

P

from the directrix of the parabola

y^2=4(x+y)

is equal to:

(A) $2$
(B) $6$
(C) $8$
(D) $4$

Q52·MathematicsSingle correctJEE Main 2023

If the tangent at a point

P

on the parabola

y^2=3x

is parallel to the line

x+2y=1

and the tangents at the points

Q

and

R

on the ellipse

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

are perpendicular to the line

x-y=2

, then the area of the triangle

PQR

is:

(A) $\dfrac32\sqrt5$
(B) $3\sqrt5$
(C) $\dfrac{9}{\sqrt5}$
(D) $5\sqrt3$

Q53·MathematicsNumericalJEE Main 2023

A triangle is formed by the tangents at the point

(2,2)

on the curves

y^2=2x

and

x^2+y^2=4x

, and the line

x+y+2=0

. If

r

is the radius of its circumcircle, then

r^2

is equal to _____.

Q54·MathematicsSingle correctJEE Main 2023

The distance of the point

(6,-2\sqrt{2})

from the common tangent

y=mx+c

,

m>0

, of the curves

x=2y^2

and

x=1+y^2

is:

(A) $\dfrac{14}{3}$
(B) $5\sqrt{3}$
(C) $\dfrac{1}{3}$
(D) $5$

Q55·MathematicsSingle correctJEE Main 2023

Let a tangent to the curve

y^2=24x

meet the curve

xy=2

at the points

A

and

B

. Then the mid points of such line segments

AB

lie on a parabola with the

(A)Length of latus rectum $\tfrac{3}{2}$
(B)directrix $4x=-3$
(C)length of latus rectum $2$
(D)directrix $4x=3$

Q56·MathematicsMultiple correctJEE Advanced 2022

Consider the parabola

y^{2} = 4x

. Let

S

be the focus of the parabola. A pair of tangents drawn to the parabola from the point

P = \left(-2, 1\right)

meet the parabola at

P_{1}

and

P_{2}

. Let

Q_{1}

and

Q_{2}

be points on the lines

SP_{1}

and

SP_{2}

respectively such that

PQ_{1}

is perpendicular to

SP_{1}

and

PQ_{2}

is perpendicular to

SP_{2}

. Then, which of the following is/are TRUE?

(A) $SQ_{1} = 2$
(B) $Q_{1}Q_{2} = \frac{3\sqrt{10}}{5}$
(C) $PQ_{1} = 3$
(D) $SQ_{2} = 1$

Q57·MathematicsSingle correctJEE Main 2022

Let the focal chord of the parabola P :

y^{2}=4x

along the line L :

y = mx + c

,

m > 0

meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H :

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

. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :

(A) $2\sqrt{6}$
(B) $2\sqrt{14}$
(C) $4\sqrt{6}$
(D) $4\sqrt{14}$

Q58·MathematicsSingle correctJEE Main 2022

If the tangents drawn at the points P and Q on the parabola

y^{2} = 2x - 3

intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :

(A)(0, 1)
(B)(2, −1)
(C)(6, 3)
(D)(2, 1)

Q59·MathematicsNumericalJEE Main 2022

Two tangent lines

l_{1}

and

l_{2}

are drawn from the point (2, 0) to the parabola

2y^{2} = -x

. If the lines

l_{1}

and

l_{2}

are also tangent to the circle

(x - 5)^{2} + y^{2} = r

, then

17r

is equal to

Q60·MathematicsSingle correctJEE Main 2022

Let P

(a, b)

be a point on the parabola

y^2 = 8x

such that the tangent at P passes through the centre of the circle

x^2 + y^2 - 10x - 14y + 65 = 0

. Let A be the product of all possible values of

a

and

B

be the product of all possible values of

b

. Then the value of A + B is equal to :

(A)0
(B)25
(C)40
(D)65

Q61·MathematicsSingle correctJEE Main 2022

If the length of the latus rectum of a parabola, whose focus is

(a, a)

and the tangent at its vertex is

x + y = a

, is 16, then

|a|

is equal to :

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

Q62·MathematicsSingle correctJEE Main 2022

The equation of a common tangent to the parabolas

y = x^{2}

and

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

is

(A) $y = 4(x-2)$
(B) $y = 4 (x-1)$
(C) $y = 4 (x+1)$
(D) $y = 4 (x+2)$

Q63·MathematicsNumericalJEE Main 2022

Let the function f(x) = 2x

^{2}

− log

_{e}

x, x > 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y

^{2}

= 4ax at a point P on it passes through the point (8a, 8a − 1) but does not pass through the point

\left(-\frac{1}{a}, 0\right)

. If the equation of the normal at P is

\frac{x}{\alpha} + \frac{y}{\beta} = 1

, then α + β is equal to-

Q64·MathematicsSingle correctJEE Main 2022

Let

P : y^{2} = 4ax

,

a > 0

be a parabola with focus S.Let the tangents to the parabola P make an angle of

\frac{\pi}{4}

with the line

y = 3x + 5

touch the parabola P at A and B. Then the value of

a

for which A,B and S are collinear is:

(A) $8$ only
(B) $2$ only
(C) $\frac{1}{4}$ only
(D)any $a > 0$

Q65·MathematicsSingle correctJEE Main 2022

If vertex of a parabola is

(2, -1)

and the equation of its directrix is

4x - 3y = 21

, then the length of its latus rectum is

(A)2
(B)8
(C)12
(D)16

Q66·MathematicsNumericalJEE Main 2022

A circle of radius 2 unit passes through the vertex and the focus of the parabola

y^2 = 2x

and touches the parabola

y=\left(x-\frac{1}{4}\right)^2+\alpha

, where

\alpha > 0

. Then

\left(4\alpha-8\right)^2

is equal to __________.

Q67·MathematicsSingle correctJEE Main 2022

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is

3x+y-29=0

, is

x^{2}+ay^{2}+bxy+cx+dy+k=0

then

a+b+c+d+k

is equal to

(A) $575$
(B) $-575$
(C) $576$
(D) $-576$

Q68·MathematicsSingle correctJEE Main 2022

Let the normal at the point P on the parabola

y^{2} = 6x

pass through the point (5, -8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

(A)-3
(B) $-\frac{9}{4}$
(C) $-\frac{5}{2}$
(D)-2

Q69·MathematicsSingle correctJEE Main 2022

If

y = m_1x + c_1

and

y = m_2x + c_2

,

m_1 \neq m_2

are two common tangents of circle

x^2 + y^2 = 2

and parabola

y^2 = x

, then the value of

8|m_1m_2|

is equal to

(A) $3+4\sqrt{2}$
(B) $-5+6\sqrt{2}$
(C) $-4+3\sqrt{2}$
(D) $7+6\sqrt{2}$

Q70·MathematicsSingle correctJEE Main 2022

Let

x = 2t

,

y = \frac{t^2}{3}

be a conic. Let S be the focus and B be the point on the axis of the conic such that SA ⊥ BA, where A is any point on the conic. If k is the ordinate of the centroid of ΔSAB, then

\lim_{t \to 1} k

is equal to

(A) $\frac{17}{18}$
(B) $\frac{19}{18}$
(C) $\frac{11}{18}$
(D) $\frac{13}{18}$

Q71·MathematicsSingle correctJEE Main 2022

If the line

y = 4 + kx

,

k > 0

, is the tangent to the parabola

y = x - x^{2}

at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

(A) $\frac{3}{2}$
(B) $\frac{26}{9}$
(C) $\frac{5}{2}$
(D) $\frac{23}{6}$

Q72·MathematicsNumericalJEE Main 2022

Let

P_{1}

be a parabola with vertex

(3, 2)

and focus

(4, 4)

and

P_{2}

be its mirror image with respect to the line

x + 2y = 6

. Then the directrix of

P_{2}

is

x + 2y =

______.

Q73·MathematicsNumericalJEE Main 2022

If two tangents drawn from a point

(\alpha,\beta)

lying on the ellipse

25x^{2}+4y^{2}=1

to the parabola

y^{2}=4x

are such that the slope of one tangent is four times the other, then the value of

\left(10\alpha+5\right)^{2}+\left(16\beta^{2}+50\right)^{2}

equals ______

Q74·MathematicsMultiple correctJEE Advanced 2021

Let E denote the parabola

y^2 = 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'

Q75·MathematicsSingle correctJEE Main 2021

Consider the parabola with vertex

\left(\frac{1}{2}, \frac{3}{4}\right)

and the directrix

y = \frac{1}{2}

. Let P be the point where the parabola meets the line

x = -\frac{1}{2}

. If the normal to the parabola at P intersects the parabola again at the point Q, then

(PQ)^{2}

is equal to :

(A) $\frac{75}{8}$
(B) $\frac{125}{16}$
(C) $\frac{25}{2}$
(D) $\frac{15}{2}$

Q76·MathematicsSingle correctJEE Main 2021

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S

(> R)

respectively from the origin, is :

(A) $4(S + R)$
(B) $2(S - R)$
(C) $4(S - R)$
(D) $2(S + R)$

Q77·MathematicsNumericalJEE Main 2021

A tangent line L is drawn at the point (2, -4) on the parabola

y^{2} = 8x

. If the line L is also tangent to the circle

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

, then 'a' is equal to ______.

Q78·MathematicsSingle correctJEE Main 2021

If two tangents drawn from a point P to the parabola

y^{2} = 16(x - 3)

are at right angles, then the locus of point P is :

(A)x + 3 = 0
(B)x + 1 = 0
(C)x + 2 = 0
(D)x + 4 = 0

Q79·MathematicsSingle correctJEE Main 2021

A tangent and a normal are drawn at the point

P(2, -4)

on the parabola

y^2 = 8x

, which meet the directrix of the parabola at the points A and B respectively. If

Q(a, b)

is a point such that AQBP is a square, then

2a + b

is equal to :

(A) $-16$
(B) $-18$
(C) $-12$
(D) $-20$

Q80·MathematicsSingle correctJEE Main 2021

Let a parabola P be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of ΔSOR is equal to :

(A)32
(B) $8\sqrt{2}$
(C)16
(D) $16\sqrt{2}$

Q81·MathematicsSingle correctJEE Main 2021

Let P be a variable point on the parabola

y=4x^{2}+1

. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line

y=x

is :

(A) $2(x-3y)^{2}+(3x-y)+2=0$
(B) $(3x-y)^{2}+(x-3y)+2=0$
(C) $2(3x-y)^{2}+(x-3y)+2=0$
(D) $(3x-y)^{2}+2(x-3y)+2=0$

Q82·MathematicsNumericalJEE Main 2021

Let

y = mx + c

,

m > 0

be the focal chord of

y^2 = -64x

, which is tangent to

\left(x + 10\right)^2 + y^2 = 4

. Then, the value of

4\sqrt{2}\left(m + c\right)

is equal to .......

Q83·MathematicsSingle correctJEE Main 2021

Let the tangent to the parabola S :

y^2 = 2x

at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

(A) $\frac{35}{2}$
(B) $\frac{15}{2}$
(C)25
(D) $\frac{25}{2}$

Q84·MathematicsSingle correctJEE Main 2021

If the three normals drawn to the parabola,

y^{2} = 2x

pass through the point (a, 0) a ≠ 0, then 'a' must be greater than :

(A) $\frac{1}{2}$
(B) $-\frac{1}{2}$
(C)−1
(D)1

Q85·MathematicsSingle correctJEE Main 2021

Let C be the locus of the mirror image of a point on the parabola

y^{2}=4x

with respect to the line

y = x

. Then the equation of tangent to C at P(2,1) is :

(A) $x-y=1$
(B) $2x+y=5$
(C) $x+3y=5$
(D) $x+2y=4$

Q86·MathematicsNumericalJEE Main 2021

A line is a common tangent to the circle (x – 3)

^{2}

+y

^{2}

=9and the parabola y

^{2}

= 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a+c) is equal to ______.

Q87·MathematicsSingle correctJEE Main 2021

The shortest distance between the line x – y = 1 and the curve x

^{2}

= 2y is:

(A)12
(B)0
(C)1 2 2
(D)1 2

Q88·MathematicsSingle correctJEE Main 2021

A tangent is drawn to the parabola

y^{2} = 6x

which is perpendicular to the line

2x + y = 1

. Which of the following points does NOT lie on it ?

(A) $(0, 3)$
(B) $(-6, 0)$
(C) $(4, 5)$
(D) $(5, 4)$

Q89·MathematicsSingle correctJEE Main 2021

The locus of the mid-point of the line segment joining the focus of the parabola

y^2=4ax

to a moving point of the parabola, is another parabola whose directrix is:.

(A) $x = a$
(B) $x = 0$
(C) $x = -\frac{a}{2}$
(D) $x = \frac{a}{2}$

Q90·MathematicsSingle correctJEE Main 2020

Let

L_1

be a tangent to the parabola

y^2 = 4(x+1)

and

L_2

be a tangent to the parabola

y^2 = 8(x+2)

such that

L_1

and

L_2

intersect at right angles. Then

L_1

and

L_2

meet on the straight line:

(A) $x+3=0$
(B) $2x+1=0$
(C) $x+2=0$
(D) $x+2y=0$

Q91·MathematicsSingle correctJEE Main 2020

If the common tangent to the parabolas

y^{2} = 4x

and

x^{2} = 4y

also touches the circle,

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

, then c is equal to

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

Q92·MathematicsSingle correctJEE Main 2020

Let P be a point on the parabola,

y^{2}=12x

and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is

\frac{4}{3}

, then:

(A) $MQ=\frac{1}{3}$
(B) $MQ=\frac{1}{4}$
(C) $PN=4$
(D) $PN=3$

Q93·MathematicsSingle correctJEE Main 2020

If one end of a focal chord AB of the parabola

y^{2} = 8x

is at

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

, then the equation of the tangent to it at B is:

(A) $2x - y - 24 = 0$
(B) $x - 2y + 8 = 0$
(C) $2x + y - 24 = 0$
(D) $x + 2y + 8 = 0$

Q94·MathematicsNumericalJEE Main 2020

Let aline

y=mx

(m>0)

intersect the parabola,

y^{2}=x

at a point P, other than the origin. Let the tangent to it at P meet the x - axis at the point Q. If area

\left(\Delta OPQ\right)=4

sq. units, then m is equal to ___________.

Q95·MathematicsSingle correctJEE Main 2020

The locus of a point which divides the line segment joining the point

(0,-1)

and a point on the parabola,

x^{2}=4y

, internally in the ratio

1:2

, is

(A) $9x^{2}-12y=8$
(B) $4x^{2}-3y=2$
(C) $x^{2}-3y=2$
(D) $9x^{2}-3y=2$

Q96·MathematicsSingle correctJEE Main 2020

If

y=mx+4

is a tangent to both the parabolas,

y^{2}=4x

and

x^{2}=2by

, then b is equal to:

(A)−32
(B)−128
(C)−64
(D)128

Q97·MathematicsSingle correctJEE Main 2019

The equation of a common tangent to the curves,

y^{2} = 16x

and

xy = -4

is :

(A) $x - 2y + 16 = 0$
(B) $2x - y + 2 = 0$
(C) $x + y + 4 = 0$
(D) $x - y + 4 = 0$

Q98·MathematicsSingle correctJEE Main 2019

If the line ax + y = c, touches both the curves

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

and

y^{2} - 4\sqrt{2}\,x

, then |c| is equal to

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

Q99·MathematicsSingle correctJEE Main 2019

If one end of a focal chord of the parabola,

y^{2}=16x

is at

(1, 4)

, then the length of this focal chord is:

(A)25
(B)24
(C)22
(D)20

Q100·MathematicsSingle correctJEE Main 2019

If the tangent to the parabola

y^{2} = x

at a point

(\alpha,\beta)

,

(\beta > 0)

is also a tangent to the ellipse,

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

, then

\alpha

is equal to

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

Q101·MathematicsSingle correctJEE Main 2019

The tangent to the parabola

y^{2}=4x

at the point where it intersects the circle

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

in the first quadrant, passes through the point:

(A) $\left(-\frac{1}{3},\frac{4}{3}\right)$
(B) $\left(\frac{3}{4},\frac{7}{4}\right)$
(C) $\left(-\frac{1}{4},\frac{1}{2}\right)$
(D) $\left(\frac{1}{4},\frac{3}{4}\right)$

Q102·MathematicsSingle correctJEE Main 2019

Let P(4, -4) and Q(9, 6) be two points on the parabola,

y^{2} = 4x

and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of

\triangle PXQ

is maximum. Then this maximum Area (in sq. units) is:

(A) $\dfrac{75}{2}$
(B) $\dfrac{125}{4}$
(C) $\dfrac{625}{4}$
(D) $\dfrac{125}{2}$

Q103·MathematicsSingle correctJEE Main 2019

The equation of a tangent to the parabola,

x^{2} = 8y

, which makes an angle

\theta

with the positive direction of x-axis, is :

(A) $y = x\tan\theta + 2\cot\theta$
(B) $y = x\tan\theta - 2\cot\theta$
(C) $x = y\cot\theta + 2\tan\theta$
(D) $x = y\cot\theta - 2\tan\theta$

Q104·MathematicsSingle correctJEE Main 2019

Equation of a common tangent to the parabola

y^{2}=4x

and the hyperbola

xy=2

is:

(A) $x+y+1=0$
(B) $x-2y+4=0$
(C) $x+2y+4=0$
(D) $4x+2y+1=0$

Q105·MathematicsSingle correctJEE Main 2019

If the area of the triangle whose one vertex is at the vertex of the parabola,

y^{2}+4\left(x-a^{2}\right)=0

and the other two vertices are the points of intersection of the parabola and y – axis, is 250 sq. units, then a value of ‘a’ is:

(A) $5\sqrt{5}$
(B) $5\left(2^{1/3}\right)$
(C) $(10)^{2/3}$
(D)5

Q106·MathematicsSingle correctJEE Main 2019

If the parabolas

y^{2}=4b(x-c)

and

y^{2}=8ax

have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?

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

Q107·MathematicsSingle correctJEE Main 2019

Axis of a parabola lies along x - axis. If its vertex and focus are at distance 2 and 4 respectively from the origin, on the positive x - axis then which of the following points does not lie on it?

(A) $(5, 2\sqrt{6})$
(B) $(8, 6)$
(C) $(6, 4\sqrt{2})$
(D) $(4, -4)$

Q108·MathematicsSingle correctJEE Main 2019

Equation of a common tangent to the circle,

x^2 + y^2 - 6x = 0

and the parabola,

y^2 = 4x

, is:

(A) $2\sqrt{3}y = 12x + 1$
(B) $\sqrt{3}y = x + 3$
(C) $2\sqrt{3}y = -x - 12$
(D) $\sqrt{3}y = 3x + 1$

Q109·MathematicsSingle correctJEE Main 2019

Let

A(4, -4)

and

B(9, 6)

be points on the parabola

y^{2} = 4x

. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of

\triangle ACB

is maximum. Then, the area (in sq. units) of

\triangle ACB

, is:

(A) $31\dfrac{3}{4}$
(B)32
(C) $30\dfrac{1}{2}$
(D) $31\dfrac{1}{4}$

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. 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 ?

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)(III) (i) (P)
(B)(III) (ii) (Q)
(C)(II) (iv) (R)
(D)(I) (ii) (Q)

Q111·MathematicsMultiple correctJEE Advanced 2017

If a chord, which is not a tangent, of the parabola

y^{2} = 16x

has the equation

2x + y = p

, and midpoint

(h, k)

, then which of the following is(are) possible value(s) of

p

,

h

and

k

?

(A) $p = 5$ , $h = 4$ , $k = -3$
(B) $p = -1$ , $h = 1$ , $k = -3$
(C) $p = -2$ , $h = 2$ , $k = -4$
(D) $p = 2$ , $h = 3$ , $k = -4$

Q112·MathematicsMultiple correctJEE Advanced 2016

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 $\frac{1}{2}$

Q113·MathematicsIntegerJEE Advanced 2015

Let the curve C be the mirror image of the parabola

y^{2} = 4x

with respect to the line

x + y + 4 = 0

. If A and B are the points of intersection of C with the line

y = -5

, then the distance between A and B is

Q114·MathematicsIntegerJEE Advanced 2015

If the normals of the parabola

y^{2} = 4x

drawn at the end points of its latus rectum are tangents to the circle

(x - 3)^{2} + (y + 2)^{2} = r^{2}

, then the value of

r^{2}

is

Q115·MathematicsMultiple correctJEE Advanced 2015

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

Q116·MathematicsIntegerJEE Advanced 2015

Suppose that the foci of the ellipse

\dfrac{x^{2}}{9} + \dfrac{y^{2}}{5} = 1

are

(f_{1}, 0)

and

(f_{2}, 0)

where

f_{1} > 0

and

f_{2} < 0

. Let

P_{1}

and

P_{2}

be two parabolas with a common vertex at

(0, 0)

and with foci at

(f_{1}, 0)

and

(2f_{2}, 0)

, respectively. Let

T_{1}

be a tangent to

P_{1}

which passes through

(2f_{2}, 0)

and

T_{2}

be a tangent to

P_{2}

which passes through

(f_{1}, 0)

. The

m_{1}

is the slope of

T_{1}

and

m_{2}

is the slope of

T_{2}

, then the value of

\left(\dfrac{1}{m_{1}^{2}} + m_{2}^{2}\right)

is

Q117·MathematicsSingle correctJEE Advanced 2014

Let

a, r, s, t

be non-zero real numbers. Let

P(at^{2}, 2at)

,

Q

,

R(ar^{2}, 2ar)

and

S(as^{2}, 2as)

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) $-\frac{1}{t}$
(B) $\frac{t^{2}+1}{t}$
(C) $\frac{1}{t}$
(D) $\frac{t^{2}-1}{t}$

Q118·MathematicsSingle correctJEE Advanced 2014

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

Q119·MathematicsSingle correctJEE Advanced 2014

Let

a, r, s, t

be non-zero real numbers. Let

P(at^{2}, 2at)

,

Q

,

R(ar^{2}, 2ar)

and

S(as^{2}, 2as)

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) $\frac{(t^{2}+1)^{2}}{2t^{3}}$
(B) $\frac{a(t^{2}+1)^{2}}{2t^{3}}$
(C) $\frac{a(t^{2}+1)^{2}}{t^{3}}$
(D) $\frac{a(t^{2}+2)^{2}}{t^{3}}$

Q120·MathematicsSingle correctJEE Advanced 2013

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) $3a$

Q121·MathematicsSingle correctJEE Advanced 2013

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

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