Parabola — JEE Maths Practice Questions & Solutions

hardPYQ · JEE 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_0

. Then three times the length of the latus rectum of

C_0

is

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mediumPYQ · JEE Main 2026

Let the directrix of

y^2=8x

cut the

x

-axis at

A

. Let

B(\alpha,\beta)

,

\alpha>1

, be on the parabola such that the slope of

AB

is

\tfrac{3}{5}

. If

BC

is a focal chord of the parabola, then six times the area of

\triangle ABC

is:

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mediumPYQ · JEE Main 2026

Let

P

be the vertex of the parabola

y = x^2-6x+12

. If a line through

P

intersects the circle

x^2+y^2-2x-4y+3=0

at points

R

and

S

, then the maximum value of

(PR+PS)^2

is:

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mediumPYQ · JEE Main 2026

Let the parabola

y=x^2+px+q

pass through the point

(1,-1)

such that the distance between its vertex and the

x

-axis is minimum. Then the value of

p^2+q^2

is

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mediumPYQ · JEE Main 2026

Consider the parabola

P:y^2=4kx

and the ellipse

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

. Let the line segment joining the points of intersection of

P

and

E

be their latus rectums. If the eccentricity of

E

is

e

, then

e^2+2\sqrt2

is equal to

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mediumPYQ · JEE Main 2026

Let chord

PQ

of length

3\sqrt{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

\alpha

at the focus of the parabola, then

\sin\alpha

is equal to

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easyPYQ · JEE Main 2026

Let

P

be a moving point on the circle

x^2+y^2-6x-8y+21=0

. Then the maximum distance of

P

from the vertex of the parabola

x^2+6x+y+13=0

is

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hard

The parabola

y = x^{2}

is reflected in the line

x + y = 1

, and the image is

x + y^{2} = ky

. Find

k

.

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medium

The length of the latus rectum of the parabola

x = t^2 + t + 1

,

y = t^2 + 2t + 3

is:

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medium

If any point

P(x, y)

which satisfies the relation

(5x - 1)^2 + (5y - 2)^2 = \lambda(3x - 4y - 1)^2

represents a parabola, then:

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medium

The locus of the vertex of the family of parabolas

y = \dfrac{a^3 x^2}{3} + \dfrac{a^2 x}{2} - 2a

, where

a

is a parameter, is:

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medium

The equation of the circle of minimum radius which touches both the parabolas

y = x^2 + 2x + 4

and

x = y^2 + 2y + 4

is:

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medium

The locus of the centre of the circle described on any focal chord of the parabola

y^2 = 4ax

as diameter is:

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medium

If the area of the triangle inscribed in the parabola

y^2 = 4ax

with one vertex at the vertex of the parabola and the other two vertices at the extremities of a focal chord is

\dfrac{5a^2}{2}

, then the length of the focal chord is:

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medium

If the tangents at the extremities of a focal chord of the parabola

x^2 = 4ay

meet the tangent at the vertex at points whose abscissae are

x_1

and

x_2

, then

x_1 x_2

is:

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medium

The area of a trapezium whose vertices lie on the parabola

y^2 = 4x

, with its diagonals passing through

(1, 0)

and having length

\dfrac{25}{4}

units each, is:

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medium

If the parabola

y = ax^2 + bx + c

has vertex at

(4, 2)

and

a \in [1, 3]

, then the difference between the extreme values of

abc

is:

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medium

Let the line

lx + my = 1

intersect the parabola

y^2 = 4ax

at points

P

and

Q

. The normals at

P

and

Q

meet at the point

R

. If

R

lies on the parabola

y^2 = 4ax

, then

a

equals:

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medium

A circle is described whose centre and diameter are the vertex and three quarters of the latus rectum of the parabola

y^2 = 4ax

, respectively. The common chord of the circle and the parabola divides the distance between the vertex and the focus in the ratio of:

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medium

Tangents are drawn to

y^2 = 4ax

from a variable point

P

moving on

x + a = 0

. Then the locus of the foot of the perpendicular drawn from

P

on the chord of contact of

P

is:

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medium

If

BC

is a latus rectum of the parabola

y^2 = 4ax

and

A

is the vertex, then the minimum length of the projection of

BC

on a tangent drawn in the portion

BAC

is:

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medium

Two distinct chords of the parabola

y^2 = 4ax

passing through

(a, 2a)

are bisected by the line

x + y = 1

. The length of the latus rectum of the parabola can be:

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medium

Let

A(0, 2)

,

B

and

C

be points on the parabola

y^2 = x + 4

such that

\angle CBA = \dfrac{\pi}{2}

. Then the range of the ordinate of

C

is:

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medium

The coordinates of the point on the parabola

y = x^2 + 7x + 2

which is nearest to the straight line

y = 3x - 3

are:

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medium

The straight line joining any point

P

on the parabola

y^2 = 4ax

to the vertex, and the perpendicular from the focus to the tangent at

P

, intersect at

R

. Then the equation of the locus of

R

is:

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medium

The locus of the point of intersection of any tangent to the parabola

y^2 = 4a(x - 2)

with a line perpendicular to it and passing through the focus is:

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medium

The locus of the focus of the family of parabolas having directrix of slope

m

and touching the lines

x = a

and

y = b

is:

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medium

A parabola

y = f(x)

has a directrix of slope

0

. If

f^{2}(0) + f^{2}(1) + f^{2}(2) + 45 = 10f(0) + 8f(1) + 4f(2),

then

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medium

The line

9x - 4y = 13

cuts a chord on the ellipse

\dfrac{x^{2}}{4} + \dfrac{y^{2}}{9} = 1

, and the mid point of that chord is

(\alpha, \beta)

. Find

(\alpha +\beta)

.

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medium

Given that

\left(3m_i^{2},\,-6m_i\right)

are the feet of the normals drawn to the parabola

y^{2} = 12x

from

(1,2)

, find the value of

\displaystyle\sum \frac{1}{m_i}

.

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medium

The two parabolas

y = x^{2}

and

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

have a common tangent. Find its equation.

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medium

Two perpendicular tangents to the parabola

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

meet at a point. Find the locus of that point.

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medium

A point moves on the parabola

y^{2} = 8x

and is joined to the focus of that parabola. Find the equation of the locus of the mid point of this segment.

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medium

A parabola is given by

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

. Then

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medium

Two tangents are drawn from a variable point

P

to the parabola

y^{2} = 4ax

, and the points where they meet the co-ordinate axes are concyclic. Given that the locus of

P

is

x = \lambda_1 a

, find

\lambda_1

.

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easy

The Cartesian equation of the curve whose parametric equations are

x = t^2 + 2t + 3

and

y = t + 1

is:

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easy

If

x + a = 0

is the directrix of the parabola

y^2 = 2y + ax + 2

, then the value of

a

is:

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easy

The locus of the centroid of the triangle formed by a tangent to the parabola

y^2 = 36x

with the coordinate axes is:

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easy

If

a \neq 0

and the line

2bx + 3cy + 4d = 0

passes through the points of intersection of the parabolas

y^2 = 4ax

and

x^2 = 4ay

, then:

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easy

If the points

(2, 3)

and

(3, 2)

on a parabola are equidistant from the focus, then the tangent at the vertex is:

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easy

Which of the following parametric equations does not represent a parabola?

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easy

The maximum number of common normals of

y^2 = 4ax

and

x^2 = 4by

may be equal to:

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easy

A parabola

y = ax^2 + bx + c

crosses the

x

-axis at

(\alpha, 0)

and

(\beta, 0)

, both to the right of the origin. A circle passes through these two points. The length of the tangent from the origin to the circle is:

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easy

The focus of the parabola

x^2 + y^2 + 2xy - 6x - 2y + 3 = 0

is:

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easy

The area of the triangle formed by the tangents at the points

(4, 6)

,

(10, 8)

and

(2, 4)

on the parabola

y^2 - 2x = 8y - 20

is (in sq. units):

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easy

A line from

(-1, 0)

intersects the parabola

x^2 = 4y

at

A

and

B

. Then the locus of the centroid of

\triangle OAB

is:

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easy

The line

x - b + \lambda y = 0

cuts the parabola

y^2 = 4ax

(a > 0)

at

P(t_1)

and

Q(t_2)

. If

b \in [2a, 4a]

, then the range of

t_1 t_2

, where

\lambda \in \mathbb{R}

, is:

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easy

Consider the parabola

x^2 + 4y = 0

. Let

P(a, b)

be a fixed point inside the parabola and let

S

be the focus. Then the minimum value of

SQ + PQ

, as the point

Q

moves on the parabola, is:

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easy

The normals are drawn from

(2k, 0)

to the curve

y^2 = 4x

. One normal is the

x

-axis. Then the value of

k

for which the other two normals are perpendicular to each other is:

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easy

The point

P

on the parabola

y^2 = 4ax

for which

|PR - PQ|

is maximum, where

R = (-a, 0)

and

Q = (0, a)

, is:

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easy

If the parabola

y = (a - b)x^2 + (b - c)x + (c - a)

touches the

x

-axis, then the line

ax + by + c = 0

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easy

If

P(-3, 2)

is one end of the focal chord

PQ

of the parabola

y^2 + 4x + 4y = 0

, then the slope of the normal at

Q

is:

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easy

If at

x = 1

the line

y = 2x

is the tangent to the parabola

y = ax^2 + bx + c

, then the respective possible values of

a

,

b

and

c

are:

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easy

Q

and

R

are the points at which the tangents drawn from the point

P(2,3)

touch the parabola

y^{2} = 4x

. Find these two points.

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Parabola — JEE Maths Practice Questions & Solutions

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