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Question

If CF be the perpendicular from the centre C of the ellipse x212+y28=1, on the tangent at any point P and G is the point where the normal at P meets the major axis, then the value of CFPG=

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Solution

Let P be (23cosθ,22sinθ)
Tangent at P:
xcosθ23+ysinθ22=1

Normal at P:
23xcosθ22ysinθ=4

The point where normal meets major axis is,
x=4cosθ23=2cosθ3
G(2cosθ3,0)

Now, CF= Distance of (0,0) from the tangent line
=|0+01| (cosθ23)2+(sinθ22)2
CF=262+sin2θ

Also, PG= (2cosθ(313))2+(22sinθ)2
=22(2+sin2θ)3

CFPG=8

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