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 D12x2+14y2=1
Tangent at R(√2sinθ,sinθ) ⇒x√2cosθ2+ysinθ1=1 A=(√2cosθ,0),B=(0,1sinθ) Let p(h,k) be the locus of the midpoint. ∴(h,k)=(√22cosθ,12sinθ)∴h=1√2cosθ,k=12sinθ⇒cosθ=1√2h,sinθ=12k ⇒cos2θ+sin2θ=12h2+14k2∴Locus of(h,k)is12x2+14y2=1