This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Conic Sections – Hyperbola”.

1. A hyperbola has ___________ vertices and ____________ foci.

a) two, one

b) one, one

c) one, two

d) two, two

View Answer

Explanation: A hyperbola has two vertices lying on each end and two foci lying inside the hyperbola.

If P is a point on hyperbola and F

_{1}and F

_{2}are foci then |PF

_{1}-PF

_{2}| remains constant.

2. The center of hyperbola is the same as a vertex.

a) True

b) False

View Answer

Explanation: No, center and vertex are different for hyperbola.

Hyperbola has one center and two vertices.

3. Find the coordinates of foci of hyperbola \((\frac{x}{9})^2-(\frac{y}{16})^2\)=1.

a) (±5,0)

b) (±4,0)

c) (0,±5)

d) (0,±4)

View Answer

Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

For hyperbola, c

^{2}=a

^{2}+b

^{2}=9+16=25 => c=5.

So, coordinates of foci are (±c,0) i.e. (±5,0).

4. Find the coordinates of foci of hyperbola \((\frac{y}{16})^2-(\frac{x}{9})^2\)=1.

a) (±5,0)

b) (±4,0)

c) (0,±5)

d) (0,±4)

View Answer

Explanation: Comparing the equation with \((\frac{y}{a})^2-(\frac{x}{b})^2\)=1, we get a=4 and b=3.

For hyperbola, c

^{2}=a

^{2}+b

^{2}= 16+9=25 => c=5.

So, coordinates of foci are (0,±c) i.e. (0,±5).

5. What is eccentricity for \((\frac{x}{9})^2-(\frac{y}{16})^2\)=1?

a) 2/5

b) 3/5

c) 15

d) 5/3

View Answer

Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

For hyperbola, c

^{2}=a

^{2}+b

^{2}= 9+16=25 => c=5.

We know, for hyperbola c=a*e

So, e=c/a = 5/3.

6. What is transverse axis length for hyperbola \((\frac{x}{9})^2-(\frac{y}{16})^2\)=1?

a) 5 units

b) 4 units

c) 8 units

d) 6 units

View Answer

Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

Transverse axis length = 2a = 2*3 =6 units.

7. What is conjugate axis length for hyperbola \((\frac{x}{9})^2-(\frac{y}{16})^2\)=1?

a) 5 units

b) 4 units

c) 8 units

d) 10 units

View Answer

Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

Conjugate axis length = 2b = 2*4 =8 units.

8. What is length of latus rectum for hyperbola \((\frac{x}{9})^2-(\frac{y}{16})^2\)=1?

a) 25/2

b) 32/3

c) 5/32

d) 8/5

View Answer

Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

We know, length of latus rectum = 2b

^{2}/a.

So, length of latus rectum of given hyperbola = 2*42/3 = 32/3.

9. What is equation of latus rectums of hyperbola \((\frac{x}{9})^2-(\frac{y}{16})^2\)=1?

a) x=±5

b) y=±5

c) x=±2

d) y=±2

View Answer

Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

For hyperbola, c

^{2}=a

^{2}+b

^{2}= 9+16=25 => c=5.

Equation of latus rectum x=±c i.e. x= ±5.

10. If length of transverse axis is 8 and conjugate axis is 10 and transverse axis is along x-axis then find the equation of hyperbola.

a) \((\frac{x}{4})^2-(\frac{y}{5})^2\)=1

b) \((\frac{x}{5})^2-(\frac{y}{4})^2\)=1

c) \((\frac{x}{10})^2-(\frac{y}{8})^2\)=1

d) \((\frac{x}{8})^2-(\frac{y}{10})^2\)=1

View Answer

Explanation: Given, 2a=8 => a=4 and 2b=10 => b=5.

Equation of hyperbola with transverse axis along x-axis is \((\frac{x}{4})^2-(\frac{y}{5})^2\)=1.

So, equation of given hyperbola is \((\frac{x}{4})^2-(\frac{y}{5})^2\)=1.

11. If foci of a hyperbola are (0, ±5) and length of semi transverse axis is 3 units, then find the equation of hyperbola.

a) \((\frac{x}{4})^2-(\frac{y}{3})^2\)=1

b) \((\frac{x}{3})^2-(\frac{y}{4})^2\)=1

c) \((\frac{x}{10})^2+(\frac{y}{8})^2\)=1

d) \((\frac{x}{8})^2-(\frac{y}{6})^2\)=1

View Answer

Explanation: Given, a=3 and c=5 => b

^{2}=c

^{2}-a

^{2}= 5

^{2}-3

^{2}=4

^{2}=> b=4.

Equation of hyperbola with transverse axis along y-axis is \((\frac{y}{a})^2-(\frac{x}{b})^2\)=1.

So, equation of given hyperbola is \((\frac{y}{3})^2-(\frac{x}{4})^2\)=1.

12. A hyperbola in which length of transverse and conjugate axis are equal is called _________ hyperbola.

a) isosceles

b) equilateral

c) bilateral

d) right

View Answer

Explanation: A hyperbola in which length of transverse and conjugate axis is equal is called equilateral hyperbola. In this type of hyperbola, a=b i.e. 2a=2b or length of transverse and conjugate axis are equal.

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