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Question

If a variable point P on an ellipse with eccentricity e is joined to it's focii S,S.Then locus of incentre of PSS is another ellipse whose eccentricity is

A
2e1+e
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B
2e1e
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C
e1e
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D
e1+e
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Solution

The correct option is A 2e1+e
Let x2a2+y2b2=1(a>b) be an ellipse

Let P(acosθ,bsinθ) be any point on ellipse ,
SP=aex=aaecosθ
SP=a+ex=a+aecosθ

Also, SS=2ae
If (h,k) is incentre of PSS


Then h=(2ae)acosθ+a(1ecosθ)(ae)+a(1+ecosθ)(ae)2ae+a(1ecosθ)+a(1+ecosθ)
h=aecosθ

Similarly, k=2ae(bsinθ)+0+02ae+2a
k=ebsinθe+1
cosθ=hae ,sinθ=k(e+1)eb

Required locus is
h2a2e2+k2(bee+1)2=1

Above locus clearly represents an ellipse whose eccentricity is e1

b2e2(1+e)2=a2e2(1e21)
e21=1b2a2(1+e)2
=1(1e2)(1+e)2
e1=2e1+e

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