wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If the eccentricity of the hyperbola x2a2-y2b2=1 is reciprocal to that of the ellipse x2+4y2=4.

If the hyperbola passes through the focus of the ellipse, then


A

The equation of the hyperbola is x23-y22=1

No worries! We‘ve got your back. Try BYJU‘S free classes today!
B

A focus of the hyperbola is 2,0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C

The eccentricity of the hyperbola is 53

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

The equation of the hyperbola is x2-3y2=3

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct option is D

The equation of the hyperbola is x2-3y2=3


Explanation for correct option

Step 1. Find the eccentricity of the ellipse

The general equation of ellipse is x2a2+y2b2=1

The eccentricity of the ellipse is e=1-ba2

Now comparing with the given equation of the ellipse x2+4y2=4x24+y2=1 with general equation, we have

a2=4,b2=1

a=±2,b=±1

So the eccentricity of the ellipse is e=1-ba2=1-14=32

Step 2. Find the focus of the ellipse

The focus of ellipse is ±ae,0

The focus of the given ellipse is ±2×32,0=±3,0

Step 4. Find the equation of the hyperbola

The hyperbolax2a2-y2b2=1 pass through ±3,0

3a2-0=1a=±3

Given that the eccentricity of the hyperbola is reciprocal of the eccentricity of the ellipse

So eccentricity of the hyperbola is e=23

We know that for the hyperbola be x2a2-y2b2=1

The eccentricity of the hyperbola is e=1+ba2

1+b2a2=231+b2a2=43b2a2=13b2±32=13b2=1b=±1

Therefore the equation of the hyperbola is x23-y2=1x2-3y2=3

The focus of the hyperbola is =±ae,0=±3×23,0=±2,0

Hence, options (B) and (D) are correct i.e. A focus of the hyperbola is 2,0 and The equation of the hyperbola is x2-3y2=3


flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Hyperbola and Terminologies
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon