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JEE • MATHEMATICS • PARABOLA

Parabola practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 511-540 of 586 questions

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511 Easy

511 of 586

Application +4 | -1

The equations of the normals at the ends of latus rectum of the parabola

y^2=4 a x

are given by

View answer and explanation

Correct option: A

x^2-y^2-6 a x+9 a^2=0

Option D (

y=m(x-a)-2 a m-a m^3

) is correct because it's the correct form of the normal equation adjusted for the shifted parabola.

512 Medium

512 of 586

Application +4 | -1

Line

\ell x+m y+n=0

is tangent to parabola if

View answer and explanation

Correct option: A

\ell n=a m^2

Option A (

\ell n=a m^2

) is correct because this condition ensures that the line is tangent to the parabola, derived from the tangency condition c = a/m.

513 Medium

513 of 586

Application +4 | -1

The equation of normal to the parabola at the point

\left(\frac{a}{m^2}, \frac{2 a}{m}\right)

View answer and explanation

Correct option: C

m^3 y=2 a m^2-m^2 x+a

Option C (

m^3 y=2 a m^2-m^2 x+a

) is correct because it represents the equation of the normal to the parabola at the given point after simplification.

514 Medium

514 of 586

Application +4 | -1

Angle between the pair of tangents to

y^2=4 a(x+a)

drawn from

(-2 a, 2)

View answer and explanation

Correct option: D

90^{\circ}

Option D (

90^{\circ}

) is correct because the point (-2a, 2) lies on the directrix of the parabola, implying the tangents from this point are perpendicular.

515 Easy

515 of 586

Application +4 | -1

Equation of chord of the parabola

{y^2} = 16x

whose mid point is

\left( {1,1} \right)

, is

View answer and explanation

Correct option: D

8x - y = 7

Option D (

8x - y = 7

) is correct because it is the equation of the chord of the parabola with the given midpoint, found using T = S1.

516 Easy

516 of 586

Analyzing +4 | -1

If the normals to the parabola

y^2=4 a x

at three points

P, Q

and

R

meet at

A

and

S

is the focus, then

S P \cdot S Q \cdot S R

is equal to

View answer and explanation

Correct option: C

a(S A)^2

Option C (

a(S A)^2

) is correct because it relates the product of distances from the focus to the points where normals meet at a point A to the distance SA.

517 Easy

517 of 586

Analyzing +4 | -1

A normal drawn at a point

P

on the parabola

y^2=4ax

meets the curve again at

Q

. The least distance of

Q

from the axis of the parabola, is

View answer and explanation

Correct option: C

4 \sqrt{2} a

Option C (

4 \sqrt{2} a

) is correct because it represents the minimum distance of Q from the axis, calculated using AM >= GM inequality.

518 Easy

518 of 586

Application +4 | -1

The polar of the point

(2,1)

with respect to the parabola

y^2=6 x

, is

View answer and explanation

Correct option: B

y=3 x+6

Option B (

y=3 x+6

) is correct because it is the polar equation of the point (2,1) with respect to the parabola

y^2=6x

519 Medium

519 of 586

Application +4 | -1

The focal distance of a point on the parabola

{y^2} + 16x

whose ordinate is twice the abscissa, is

View answer and explanation

Correct option: B

Option B (8) is correct because the focal distance of the point on the parabola is 8.

520 Easy

520 of 586

Application +4 | -1

The parabola with directrix

x + 2y - 1 = 0

and focus

\left( {1,0} \right)

View answer and explanation

Correct option: A

4{x^2} - 4xy + {y^2} - 8x + 4y + 4 = 0

Option A (

4{x^2} - 4xy + {y^2} - 8x + 4y + 4 = 0

) is correct because it represents the equation of the parabola with the given directrix and focus.

521 Medium

521 of 586

Application +4 | -1

Equation of a common tangent to the curves

y^2=8 x

and

x y=-1

View answer and explanation

Correct option: D

y=x+2

Option D (

y=x+2

) is correct because it represents a common tangent to both the parabola and the hyperbola, derived from the tangency conditions.

522 Medium

522 of 586

Application +4 | -1

If the line

2 x+y+k=0

is normal to the parabola

y^2=-8 x

, then the value of

k

View answer and explanation

Correct option: D

The correct answer is D (24) because substituting the normal form into the parabola equation and solving for k yields 24.

523 Medium

523 of 586

Application +4 | -1

The straight line

y=m x+c(m>0)

touches the parabolas

y^2=8(x+2)

then the minimum value taken by

c

View answer and explanation

Correct option: A

The correct answer is A (4) because the minimum value of c is found by using the tangent equation and applying AM-GM inequality.

524 Medium

524 of 586

Application +4 | -1

If tangents at

A

and

B

on the parabola

y^2=4 a x

intersect at point

C

, then ordinates of

A, C

and

B

are

View answer and explanation

Correct option: A

Always in

AP

Option A (Always in AP) is correct because the ordinate of the intersection point C is the average of the ordinates of A and B, indicating an arithmetic progression.

525 Easy

525 of 586

Analyzing +4 | -1

The equation

{y^2} - 8y-x + 19 = 0

represents a parabola whose focus is

\left( {\frac{1}{4},0} \right)

and directrix is a parabola whose vertex is (3,4) and directrix is

View answer and explanation

Correct option: B

x = \frac{{11}}{4}

Option B (

x = \frac{{11}}{4}

) is correct because it is the directrix of the parabola, calculated by finding the directrix in the translated coordinate system and then converting back to the original coordinates.

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526 Hard

526 of 586

Analyzing +4 | -1

If two different tangents of

y^2=4 x

are the normals to

x^2=4

by, then

View answer and explanation

Correct option: A

|b|<\frac{1}{2 \sqrt{2}}

Option A (

|b|<\frac{1}{2 \sqrt{2}}

) is correct because the condition for two different tangents leads to a discriminant greater than zero, resulting in the given inequality.

527 Easy

527 of 586

Application +4 | -1

If the vertex of a parabola is

\left( {0,2} \right)

and the extremities of latusrectum are

\left( { - 6,4} \right)

and

\left( {6,4} \right),

then, its equation is

View answer and explanation

Correct option: C

{x^2} - 8y + 16 = 0

The equation of the parabola is

{x^2} - 8y + 16 = 0

because it is derived from the vertex and latus rectum information.

528 Easy

528 of 586

Application +4 | -1

Equation of a common tangent to the parabola

{y^2} = 4x

and the hyperbola

xy = 2

View answer and explanation

Correct option: A

x + 2y + 4 = 0

Option A (

x + 2y + 4 = 0

) is correct because it represents the common tangent found by equating the tangent equations of the parabola and hyperbola.

529 Easy

529 of 586

Analyzing +4 | -1

The number of points on

y^2=4ax

at which the normal chords drawn subtend a right angle at the vertex, is

View answer and explanation

Correct option: B

Option B (2) is correct because there are two points on the parabola where the normal chords drawn subtend a right angle at the vertex, derived from the condition t² = 2.

530 Easy

530 of 586

Application +4 | -1

If the straight line

lx + my + n = 0

touches the parabola,

{y^2} = 4ax

then

View answer and explanation

Correct option: B

nl = a{m^2}

Option B (

nl = a{m^2}

) is correct because it represents the condition for tangency of the line

lx + my + n = 0

to the parabola

{y^2} = 4ax

531 Medium

531 of 586

Analyzing +4 | -1

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

View answer and explanation

Correct option: A

x + 3 = 0

Option A is correct because the locus of the intersection point of the perpendicular tangents to the given parabolas is the line

x + 3 = 0

532 Hard

532 of 586

Analyzing +4 | -1

a \neq 0

and the line

2 b x+3 c y+4 d=0

passes through the points of intersection of the parabolas

y^2=4 a x

and

x^2=4 a y

, then

View answer and explanation

Correct option: A

d^2+(2 b+3 c)^2=0

Option A (

d^2+(2 b+3 c)^2=0

) is correct because the given line passes through the intersection points of the parabolas, leading to the condition d=0 and 2b+3c=0.

533 Medium

533 of 586

Application +4 | -1

The points of intersection of the curves whose parametric equations are

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

and

x = 2s,y = 2/s

is given by

View answer and explanation

Correct option: B

\left( {2,2} \right)

Option B ((2,2)) is correct because it is the point that satisfies both parametric equations when eliminating the parameters t and s.

534 Easy

534 of 586

Application +4 | -1

P\left( {at_1^2,2a{t_1}} \right)

and

Q\left( {at_2^2,2a{t_2}} \right)

are two variable points on the curve

{y^2} = 4\;ax

and

PQ

subtends a right angle at the vertex, then

{t_1}{t_2}

is equal to

View answer and explanation

Correct option: D

- 4

Option D (

-4

) is correct because the product of

t_1

and

t_2

is equal to

-4

when

PQ

subtends a right angle at the vertex of the parabola.

535 Medium

535 of 586

Application +4 | -1

The set of positive value of

m

for which a line with slope

m

is a comon tangent to ellipse

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

and parabola

y^2=4 a x

is given by

View answer and explanation

Correct option: C

(0,1)

Option C ((0,1)) is correct because it represents the set of positive values of m for which a line with slope m is a common tangent to the given ellipse and parabola.

536 Easy

536 of 586

Application +4 | -1

The locus of the point of intersection of perpendicular tangents to the parabola

y^2=4 a x

, is

View answer and explanation

Correct option: B

x=-a

Option B (

x=-a

) is correct because the locus of the intersection of perpendicular tangents to a parabola is its directrix, which is x = -a for the parabola

y^2 = 4ax

537 Medium

537 of 586

Application +4 | -1

If a normal drawn to the parabola

y^2=4 a x

at the point

(a, 2 a)

meets parabola again on

\left(a t^2, 2 a t\right)

, then the value of

t

View answer and explanation

Correct option: D

-3

Option D (-3) is correct because it uses the formula relating the parameters of the points where the normal intersects the parabola.

538 Easy

538 of 586

Application +4 | -1

The length of the latusrectum of the parabola

{x^2} - 4x - 8y + 12 = 0

View answer and explanation

Correct option: C

Option C (8) is correct because it's the length of the latus rectum calculated after rewriting the parabola equation in standard form.

539 Easy

539 of 586

Application +4 | -1

The length of the common chord of the parabolas

{y^2} = x

and

{x^2} = y

and is

View answer and explanation

Correct option: C

\sqrt 2

Option C

\sqrt 2

is correct because it is the distance between the intersection points (0,0) and (1,1) of the two parabolas.

540 Medium

540 of 586

Analyzing +4 | -1

Let

A, B

, and

C

be three points taken on the parabola

y^2=4 a x

with coordinates

\left(a t_i{ }^2, 2 a t_i\right) ; i=1,2,3

; where

t_1, t_2

and,

t_3

are in A.P. If

A A^{\prime}, B B^{\prime}

and

C C^{\prime}

are focal chords and coordinates of

A^{\prime}, B^{\prime}

and

C^{\prime}

are

\left(a t_i^{\prime^2}, 2 a t_i^{\prime}\right) ; i=1,2,3

; then

t^{\prime}{ }_1, t^{\prime}{ }_2

and

t^{\prime}{ }_3

are in

View answer and explanation

Correct option: C

Option C (HP) is correct because if

t_1, t_2, t_3

are in AP, then

t'_1, t'_2, t'_3

are in HP, where

t'_i = -1/t_i

Revision notes and key takeaways

Review PARABOLA in your textbook or class notes, then use this set to check your understanding.

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Parabola practice · Mixed difficulty

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