Back to HYPERBOLA

Your Progress

0 / 118 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • HYPERBOLA

Hyperbola practice · Medium

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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.

Take a chapter quiz →

Showing 1-30 of 118 questions

1 Medium

1 of 118

Analyzing +4 | -1

Let

P\left(x_{0}, y_{0}\right)

be the point on the hyperbola

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

=36

, which is nearest to the line

3 x+2 y=1

. Then

\sqrt{2}\left(y_{0}-x_{0}\right)

is equal to

View answer and explanation

Correct option: C

-9

Option C (-9) is correct because it involves finding the point on the hyperbola where the tangent is parallel to the given line and then calculating the required expression.

2 Medium

2 of 118

Application +4 | -1

Suppose an ellipse and a hyperbola have the same pair of foci on the

x

-axis with centres at the origin and they intersect at

(2,2)

. If the eccentricity of the ellipse is

\frac{1}{2}

, then the eccentricity of the hyperbola, is

View answer and explanation

Correct option: B

\sqrt{\frac{7}{3}}

Option B (

\sqrt{\frac{7}{3}}

) is correct because it is the eccentricity of the hyperbola derived from the given conditions of shared foci and intersection point.

3 Medium

3 of 118

Analyzing +4 | -1

Study the following statement and select the correct option:

\mathrm{A}

is the vertex of a branch of the hyperbola

x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0, \mathrm{~B}

is one of the end point of its latusrectum and

\mathrm{C}

is the focus of the hyperbola nearer to the point

A

Statement-I : The area of the triangle

\mathrm{ABC}

\sqrt{\frac{3}{2}}-1

Statement-II: Eccentricity of the hyperbola is

2 \sqrt{2}

and length of the conjugate axis is 4

View answer and explanation

Correct option: C

Statement-1 is true, statement-2 is false

Option C (Statement-1 is true, statement-2 is false) is correct because the area of the triangle is indeed

\sqrt{\frac{3}{2}}-1

, but the eccentricity and length of the conjugate axis are not as stated.

4 Medium

4 of 118

Application +4 | -1

The line

l x+m y+n=0

is a tangent to the hyperbola

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

View answer and explanation

Correct option: B

a^2 l^2-b^2 m^2=n^2

Option B (

a^2 l^2-b^2 m^2=n^2

) is the correct condition for tangency of a line to a hyperbola.

5 Medium

5 of 118

Application +4 | -1

Locus of intersection of the two perpendicular tangents to the given hyperbola is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the director circle equation results in a negative radius squared, indicating no real locus.

6 Medium

6 of 118

Application +4 | -1

A tangent drawn to hyperbola

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

P\left(\frac{\pi}{6}\right)

forms a triangle of area

3 a^2

sq. units, with coordinate axes. Its eccentricity is equal to

View answer and explanation

Correct option: D

\sqrt{17}

Option D is correct because the area of the triangle formed by the tangent is used to find the relationship between a and b, which is then used to calculate the eccentricity.

7 Medium

7 of 118

Application +4 | -1

{e_1}

is the eccentricity of the ellipse

\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1

and

{e_2}

is the eccentricity of the hyperbola passing through the foci of the ellipse and

{e_1}{e_2} = 1

, then equation of the hyperbola is

View answer and explanation

Correct option: B

\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = - 1

Option B (

\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = - 1

) is correct because it satisfies the condition that the hyperbola passes through the foci of the ellipse and has the given eccentricity relationship.

8 Medium

8 of 118

Application +4 | -1

x^2+y^2=k^2

, (where

x

and

y

are variables and

k

is constant), then

x \& y

can be taken as

x=k \cos \theta

\& y=k \sin \theta

. If

x^2+y^2=9

, then minimum value of the expression

\frac{9}{x^2}+\frac{9}{y^2}

is equal to -

View answer and explanation

Correct option: A

The correct answer is A (4) because the minimum value of the expression

\frac{9}{x^2} + \frac{9}{y^2}

is found by substituting

x = 3\cos\theta

and

y = 3\sin\theta

and then using the fact that

tan^2 \theta + cot^2 \theta >= 2

9 Medium

9 of 118

Application +4 | -1

If distance between directrices of a rectangular hyperbola is 10, then distance between its foci will be

View answer and explanation

Correct option: D

Option D (20) is correct because, for a rectangular hyperbola, e = √2, and using the given distance between directrices, the distance between foci is calculated as 20.

10 Medium

10 of 118

Analyzing +4 | -1

Equation of asymptotes of

xy = 7x + 5y

are

View answer and explanation

Correct option: B

x = 5,y = 7

Option B (

x = 5, y = 7

) is correct because these are the asymptotes of the given hyperbola after rewriting the equation.

11 Medium

11 of 118

Analyzing +4 | -1

The circle

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

and hyperbola

\frac{x^2}{9}-\frac{y^2}{4}=1

intersect at the points

A

and

B

. The equation of a common tangent with positive slope to the circle as well as to the hyperbola, is

View answer and explanation

Correct option: B

2 x-\sqrt{5} y+4=0

Option B (

2 x-\sqrt{5} y+4=0

) is correct because it represents a common tangent with a positive slope to both the circle and the hyperbola.

12 Medium

12 of 118

Application +4 | -1

eccentricity of the hyperbola conjugate to the hyperbola

\frac{x^2}{4}-\frac{y^2}{12}=1

View answer and explanation

Correct option: A

2 / \sqrt{3}

Option A (

2 / \sqrt{3}

) is correct because it is the eccentricity of the hyperbola conjugate to the given hyperbola, calculated using the relationship between the eccentricities of conjugate hyperbolas.

13 Medium

13 of 118

Application +4 | -1

y=m x+6

is a tangent to the hyperbola

\frac{x^2}{100}-\frac{y^2}{49}=1

and the parabola

y^2=4 a x

, then the length of the latus rectum of the parabola is

View answer and explanation

Correct option: C

24 \sqrt{\frac{17}{20}}

Option C (

24 \sqrt{\frac{{17}}{{20}}}

) is correct because it involves finding the slope of the tangent to the hyperbola, then using that slope to find 'a' for the parabola and finally calculating the latus rectum.

14 Medium

14 of 118

Application +4 | -1

If the tangent at the point

\left( {a\sec \alpha ,b\tan \alpha } \right)

to the hyperbola

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

meets the transverse axis at

T

, then the distance of

T

from a focus of the hyperbola is

View answer and explanation

Correct option: C

a\left( {e + \cos \alpha } \right)

Option C (

a\left( {e + \cos \alpha } \right)

) is correct because it represents the distance between the point T on the transverse axis and a focus of the hyperbola.

15 Medium

15 of 118

Analyzing +4 | -1

If equation

{\left( {10x - 5} \right)^2} + {\left( {10y - 4} \right)^2} = {\lambda ^2}{\left( {3x + 4y - 1} \right)^2}

represents a hyperbola, then

View answer and explanation

Correct option: C

\lambda < - 2

\lambda > 2

Option C (

\lambda < -2

\lambda > 2

) is correct because the eccentricity of a hyperbola is greater than 1, leading to the condition

|\lambda| > 2

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

16 Medium

16 of 118

Analyzing +4 | -1

Let

\mathrm{S}=\left\{(x, y) \in \mathbf{R}^{2}: \frac{y^{2}}{1+\mathrm{r}}-\frac{x^{2}}{1-\mathrm{r}}=1\right\}

where

r \neq \pm 1

Then

S

represents:

View answer and explanation

Correct option: B

an ellipse whose eccentricity is

\sqrt{\frac{2}{r+1}}, \text { when } r > 1

Option B (an ellipse whose eccentricity is

\sqrt{{\frac{{2}}{{r+1}}}}, \text{{ when }} r > 1

) is correct because when

r > 1

, the equation represents an ellipse, and the eccentricity is calculated correctly.

17 Medium

17 of 118

Application +4 | -1

The eccentricity of the hyperbola

5 x^2-4 y^2+20 x+8 y=4

View answer and explanation

Correct option: B

3 / 2

Option B (3/2) is correct because after converting the equation to standard form, the eccentricity can be calculated using the relationship between a, b, and e.

18 Medium

18 of 118

Application +4 | -1

x = 9

is the chord of contact of the hyperbola

{x^2} - {y^2} = 9,

then the equation of the corresponding pair of tangents is

View answer and explanation

Correct option: B

9{x^2} - 8{y^2} - 18x + 9 = 0

Option B (

9{x^2} - 8{y^2} - 18x + 9 = 0

) is correct because it is derived from the condition that x=9 is the chord of contact and using the equation

T^2 = SS_1

19 Medium

19 of 118

Application +4 | -1

C

is the centre of the hyperbola

\frac{x^2}{4}-\frac{y^2}{1}=1

, and '

A

' is any point on it. The tangent at

{A}

to the hyperbola meets the line

x-2 y=0

and

x+2 y=0

Q

and

R

respectively. The value of

C Q \cdot C R

is equal to

View answer and explanation

Correct option: A

Option A (5) is correct because the product CQ * CR is calculated using the tangent equation and the given hyperbola.

20 Medium

20 of 118

Application +4 | -1

Locus of the mid-points of the chords of the hyperbola

\frac{x^2}{16}-\frac{y^2}{9}=1

that are parallel to

3 x-4 y=1

View answer and explanation

Correct option: B

3 x-4 y=0

Option B (

3x - 4y = 0

) is correct because the locus of the midpoints of chords parallel to a given line can be found using the equation of the chord with midpoint (h,k) and equating its slope to the slope of the given line.

21 Medium

21 of 118

Application +4 | -1

Foci of the hyperbola

\frac{x^2}{16}-\frac{(y-2)^2}{9}=1

are

View answer and explanation

Correct option: A

(5,2)(-5,2)

Option A ((

5,2

(-5,2)

) is correct because these are the foci calculated using the eccentricity and center of the given hyperbola.

22 Medium

22 of 118

Application +4 | -1

If the line

y = mx + 7\sqrt 3

is normal to the hyperbola

\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1,

then a value of

m

View answer and explanation

Correct option: C

\frac{2}{{\sqrt 5 }}

Option C (

\frac{2}{{\sqrt 5 }}

) is correct because it is a value of m for which the line

y = mx + 7\sqrt 3

is normal to the hyperbola

\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1

23 Medium

23 of 118

Analyzing +4 | -1

A hyperbola touches y – axis and has its centre and one of the foci are at

\left( {\frac{5}{{20}},20} \right)

and
(10, 24) respectively; then length of its transverse axis is

View answer and explanation

Correct option: C

\sqrt {89}

Option C (

\sqrt {89}

) is the correct answer because it is derived from the geometric properties of the hyperbola and the given conditions.

24 Medium

24 of 118

Application +4 | -1

Tangent at point

P

of hyperbola

x y=14

intersects the coordinate axes at

A

and

B

. The locus of centroid of

\triangle \mathrm{OAB}

where

O

is origin, is

View answer and explanation

Correct option: D

a hyperbola

Option D (a hyperbola) is correct because the locus of the centroid of the triangle formed by the tangent to the hyperbola and the coordinate axes is a hyperbola.

25 Medium

25 of 118

Application +4 | -1

The equation of the chord of contact of tangents drawn from a point

(2,-1)

to the hyperbola

16 x^2-9 y^2=144

, is

View answer and explanation

Correct option: C

32 x+9 y=144

Option C (

32 x+9 y=144

) is correct because it is the equation of the chord of contact of tangents drawn from the given point to the hyperbola.

26 Medium

26 of 118

Analyzing +4 | -1

The foci of a hyperbola are

(-5,18)

and

(10,20)

and it touches the

y

-axis. The length of its transverse axis, is

View answer and explanation

Correct option: C

\sqrt{89}

Option C (

\sqrt{89}

) is correct because it represents the length of the transverse axis of the hyperbola, calculated using the distance between the foci and the condition that the hyperbola touches the y-axis.

27 Medium

27 of 118

Application +4 | -1

The equation of the hyperbola whose vertices are

( \pm 2,0)

and foci are

( \pm 3,0)

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

. Sum of

{a}^2

and

{b}^2

View answer and explanation

Correct option: A

Option A (9) is correct because the sum of a² and b² for the given hyperbola is 9, calculated using the vertices and foci.

28 Medium

28 of 118

Application +4 | -1

P

and

Q

are two points on the rectangular hyperbola

x y=c^2

such that the abscissa of

P

and

Q

are the roots of the equations

x^2-6 x-16=0

. Equation of the chord joining

P

and

Q

View answer and explanation

Correct option: C

c^2 x-16 y=6 c^2

Option C (

c^2 x-16 y=6 c^2

) is correct because it is the equation of the chord joining P and Q on the rectangular hyperbola.

29 Medium

29 of 118

Application +4 | -1

Two tangents are drawn to the hyperbola

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

such that the product of their slopes is

c^2

. The locus of their point of intersection is

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the locus of the point of intersection of two tangents with a product of slopes equal to

c^2

is given by this equation.

30 Medium

30 of 118

Analyzing +4 | -1

If the tangent at a point

P

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

cuts one of its directrices at

Q

, then the angle made by

PQ

at the corresponding focus, is

View answer and explanation

Correct option: D

{90^0}

Option D (

{90^0}

) is correct because the tangent at a point P on the hyperbola cuts the directrix at Q, and PQ subtends a right angle at the corresponding focus.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 118 single-correct-answer MCQs on HYPERBOLA.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

Showing 1-30 of 118 questions

Continue Your Practice

📚 Switch Set (MEDIUM)

Chapter Sets

ALL 429 MCQs EASY 298 MCQs HARD 13 MCQs MEDIUM 118 MCQs

0 /118

0✓ 0✗

0% Accuracy