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Question

If Y-axis is the directrix of the ellipse with eccentricity e=1/2 and corresponding focus is at (3,0), then the equation to its auxiliary circle is

A
x2+y28x+12=0
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B
x2+y28x12=0
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C
x2+y28x+9=0
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D
x2+y2=4
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Solution

The correct option is A x2+y28x+12=0
Let center of the ellipse be (h,k)

Given that y-axis is the directrix.

x=0

We know that directrix is x=hae

So, hae=0

h=ae

h=a1/2

h=2a

We know that focus =(hae,k)

(hae,k)=(3,0)

hae=3

2a(a2)=3

3a2=3

a=2

a2=4

We know that b2=a2(1e2)

h=2a=4;k=0

b2=4(114)

b2=3

Thus equation of ellipse(x4)24+y23=1

Hence, equation of auxiliary circle is (xh)2+(yk)2=a2

(x4)2+y2=4

x2+y28x+16=4

x2+y28x+12=0

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