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Question

If the normal at the point P(θ) to the ellipse x214+y25=1 intersects it again at the point Q(2θ), then

A
sinθ=23
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B
sinθ=34
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C
cosθ=34
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D
cosθ=23
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Solution

The correct option is D cosθ=23
Equation of ellipse is
x214+y25=1
P(θ)(14cosθ,5sinθ)

The normal at P will be
14xcosθ5xsinθ=14514xcosθ5xsinθ=9

If it meets the curve again at Q(2θ)(14cos2θ,5sin2θ), then

14(14cos2θ)cosθ5(5sin2θ)sinθ=9
14(2cos2θ1)cosθ5(2cosθ)=9
28cos2θ1410cos2θ=9cosθ
18cos2θ9cosθ14=0
(6cosθ7)(3cosθ+2)=0
cosθ=23 or cosθ=76 (Not possible)
Hence, cosθ=23

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