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Question

The product of the perpendiculars drawn from the two foci of the ellipse x2a2+y2b2=1(a>b) to the tangent at any point of the ellipse is:

A
a2
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B
4a2
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C
4b2
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D
b2
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Solution

The correct option is C b2
The equation of the tangent at any point (acosθ,bsinθ) on the ellipse is

xacosθ+ybsinθ=1i.e.,bxcosθ+aysinθab=0.

The product of the perpendiculars from S(ae,0) and S(ae,0) on this tangent

=(abecosθab)(abecosθab)b2cos2θ+a2sin2θ

=a2b2(1e2cos2θ)a2(1e2)cos2θ+a2sin2θ=a2b2(1e2cos2θ)a2(1e2cos2θ)=b2

Ans: D

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