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Question

Consider a tangent to the ellipse x22+y21=1 at any point. The locus of the midpoint of the portion intercepted between the axes is

A
x22+y24=1
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B
x24+y22=1
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C
13x2+14y2=1
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D
12x2+14y2=1
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Solution

The correct option is D 12x2+14y2=1

Tangent at R (2sinθ,sinθ)
x2 cosθ2+y sinθ1=1
A=(2cosθ,0), B=(0,1sinθ)
Let p(h,k) be the locus of the midpoint.
(h,k)=(22cosθ,12sinθ)h=12 cosθ, k=12sinθcosθ=12 h, sinθ=12k
cos2θ+sin2θ=12h2+14k2 Locus of (h,k) is 12x2+14y2=1

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