Parabola29 questions

Parabola — JEE Maths Practice Questions & Solutions

29 questions on Parabola with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

The length of the latus rectum of the parabola

x = t^2 + t + 1

y = t^2 + 2t + 3

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}

y = ax^2 + bx + c

has vertex at

(4, 2)

and

a \in [1, 3]

, then the difference between the extreme values of

abc

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

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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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

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

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

(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

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

(-1, 0)

intersects the parabola

x^2 = 4y

A

and

B

. Then the locus of the centroid of

\triangle OAB

x - b + \lambda y = 0

cuts the parabola

y^2 = 4ax

(a > 0)

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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