Mathematics Questions and Answers – Conic Sections – Hyperbola

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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

Answer: d
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 F1 and F2 are foci then |PF1-PF2| remains constant.
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2. The center of hyperbola is the same as a vertex.
a) True
b) False
View Answer

Answer: b
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

Answer: a
Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.
For hyperbola, c2=a2+b2=9+16=25 => c=5.
So, coordinates of foci are (±c,0) i.e. (±5,0).
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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

Answer: c
Explanation: Comparing the equation with \((\frac{y}{a})^2-(\frac{x}{b})^2\)=1, we get a=4 and b=3.
For hyperbola, c2=a2+b2= 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

Answer: d
Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.
For hyperbola, c2=a2+b2= 9+16=25 => c=5.
We know, for hyperbola c=a*e
So, e=c/a = 5/3.
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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

Answer: d
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

Answer: c
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.
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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

Answer: b
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 = 2b2/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

Answer: a
Explanation: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.
For hyperbola, c2=a2+b2= 9+16=25 => c=5.
Equation of latus rectum x=±c i.e. x= ±5.
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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

Answer: a
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

Answer: a
Explanation: Given, a=3 and c=5 => b2=c2-a2 = 52-32=42 => 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

Answer: b
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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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter