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JEE • MATHEMATICS • PARABOLA

Parabola practice · Hard

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Practice format and chapter quiz

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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1 Hard

1 of 19

Application +4 | -1

I

denotes the semi-latusrectum of the parabola

{y^2} = 4ax

and

SP

and

SQ

denote the segments of any focal chord

PQ,S

being the focus, then

SP,l

and

SQ

are in the relation

View answer and explanation

Correct option: C

Option C (HP) is correct because the semi-latus rectum

l

, and the segments of the focal chord

SP

and

SQ

satisfy the harmonic progression relation, i.e.,

2/l = 1/SP + 1/SQ

2 Hard

2 of 19

Application +4 | -1

A common tangent is drawn to the circle

x^2+y^2=c^2

and the parabola

y^2=4 a x

. If the angle which this tangent makes with the axis of

x

\pi / 4

then the relationship between a and

c(a, c>0)

is-

View answer and explanation

Correct option: A

\mathrm{a}=\sqrt{2} \mathrm{c}

Option D (

2 x+y-3 \sqrt{{3}}=0

) is correct because it is the equation of the tangent calculated at the point where the ordinate and abscissa are equal on the curve.

3 Hard

3 of 19

Analyzing +4 | -1

The set of points on the axis of the parabola

y^2-4 x-2 y+5=0

from which all the three normals to the parabola are real is

View answer and explanation

Correct option: B

(k, 1) ; k>3

Option B (

(k, 1) ; k>3

) is correct because completing the square gives the standard form of the parabola, and the condition for three real normals from a point on the axis leads to x > 3 and y = 1.

4 Hard

4 of 19

Application +4 | -1

If the length of the normal chord to the parabola

\mathrm{y}^2=4 \mathrm{x}

, which subtends right angle at the vertex is

\mathrm{L}

then the value of

\mathrm{L}^2

View answer and explanation

Correct option: A

108

Option A (108) is correct because the length of the normal chord is calculated to be

6\sqrt{3}

, and squaring this value gives 108.

5 Hard

5 of 19

Application +4 | -1

The chord of contact of tangents drawn from any point on

x - 1 = 0

{y^2} - 6y + 4x + 9 = 0

passes through the point

View answer and explanation

Correct option: A

\left( { - 1,3} \right)

Option A ((-1,3)) is correct because the chord of contact of tangents drawn from any point on the directrix of a parabola passes through its focus.

6 Hard

6 of 19

Analyzing +4 | -1

If the 4th term in the expansion of

\left(p x+\frac{1}{x}\right)^n, n \in N

\frac{5}{2}

and three normals to the parabola

y^2=x

are drawn through a point

(q, 0)

, then

View answer and explanation

Correct option: B

q>p

Option B (

q>p

) is correct because the condition for three normals to a parabola to be drawn from a point (q,0) implies q > 2a, and we find p = 1/2 and a = 1/4, thus q > p.

7 Hard

7 of 19

Analyzing +4 | -1

Let

y=f(x)

represent a parabola with focus

\left(-\frac{1}{2}, 0\right)

and directrix

y=-\frac{1}{2}

. Then

S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}:

View answer and explanation

Correct option: D

Contains exactly two elements

Option D (Contains exactly two elements) is correct because solving the equation derived from the given conditions yields two real solutions for x.

8 Hard

8 of 19

Analyzing +4 | -1

The locus of the vertices of the family of parabolas

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

View answer and explanation

Correct option: A

x y=\frac{105}{64}

Option A (

x y=\frac{105}{64}

) is correct because finding the vertex of the parabola and eliminating the parameter 'a' leads to the given locus.

9 Hard

9 of 19

Application +4 | -1

The equation of the common tangent to the curves

{y^2} = 8x

and

xy = - 1

View answer and explanation

Correct option: D

y = x + 2

Option D (

y = x + 2

) is correct because it is the common tangent to both the parabola

{y^2} = 8x

and the hyperbola

xy = -1

, found by equating the tangent equation of the parabola to the hyperbola and solving for the tangency condition.

10 Hard

10 of 19

Analyzing +4 | -1

For each parabola

y=x^2+p x+q

, meeting coordinate axes at 3-distinct points, if circles are drawn through these points, then the family of circles must pass through

View answer and explanation

Correct option: B

(0,1)

Option B (0,1) is correct because the family of circles passing through the intersection points of the parabola and coordinate axes must pass through the point (0,1).

11 Hard

11 of 19

Understanding +4 | -1

The locus of the foot of the perpendicular from the focus upon a tangent to the parabola

{y^2} = 4\;ax

View answer and explanation

Correct option: B

Tangent at the vertex

Option B (Tangent at the vertex) is correct because the locus of the foot of the perpendicular from the focus upon a tangent to the parabola is the tangent at the vertex.

12 Hard

12 of 19

Application +4 | -1

Radius of circle touching the parabola

y^2=4 x

at point

(1,2)

and passing through its focus is

View answer and explanation

Correct option: B

\sqrt{2}

Option B (

\sqrt{2}

) is correct because it's the radius of the circle that touches the parabola at (1,2) and passes through the focus (1,0).

13 Hard

13 of 19

Analyzing +4 | -1

In a parabola

y^2 = 4ax,

two points

P

and

Q

are taken such that the tangents drawn to parabola at these points
meet at directrix in

R.

Focus of locus of circumcentre of

\Delta PQR

will be:

View answer and explanation

Correct option: C

\left( {\frac{{3a}}{2},0} \right)

Option C

\left( \frac{{3a}}{{2}},0 \right)

is correct because the locus of the circumcentre is a parabola with focus at

\left( \frac{{3a}}{{2}},0 \right)

14 Hard

14 of 19

Understanding +4 | -1

The circle on focal radii of a parabola as diameter touches the

View answer and explanation

Correct option: C

Tangent at the vertex

The correct answer is C (Tangent at the vertex) because the circle described on the focal radii of a parabola as diameter touches the tangent at the vertex.

15 Hard

15 of 19

Analyzing +4 | -1

If the normals from any point to the parabola

x^2=4 y

cuts the line

y=2

in points whose abscissae are in A.P., then the slopes of the tangents at the 3 conormal points are in

View answer and explanation

Correct option: B

G P

Option B (G P) is correct because the slopes of the tangents at the three conormal points are in Geometric Progression.

Explore Other Chapters in MATHEMATICS

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16 Hard

16 of 19

Analyzing +4 | -1

The set of points

(x, y)

whose distance from the line

y=2 x+2

is the same as the distance from

(2,0)

is a parabola. This parabola is congruent to the parabola in standard from

y=k x^2

for some

k

which is equal to

View answer and explanation

Correct option: D

\frac{\sqrt{5}}{12}

Option D (

\frac{\sqrt{5}}{12}

) is correct because equating the distance from a point to a line and a fixed point results in a parabola, and transforming it to standard form reveals k =

\frac{\sqrt{5}}{12}

17 Hard

17 of 19

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.

18 Hard

18 of 19

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.

19 Hard

19 of 19

Application +4 | -1

y = 2x + 3

is a tangent to the parabola,

{y^2} = 24\;x,

then its distance from the parallel normal is

View answer and explanation

Correct option: C

15\sqrt 5

Option C (

15\sqrt 5

) is correct because it is the distance between the given tangent and its parallel normal to the parabola.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 19 single-correct-answer MCQs on PARABOLA.
✓ Try each question before opening its answer. Read the explanation where one is available.
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