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Question

the locus of a midpoint of a chord of the circle x2+y2=4 which subtends a right angle at the origin is

A
x+y=2
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B
x2+y2=1
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C
x2+y2=2
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D
x+y=1
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Solution

The correct option is C x2+y2=2
Let the end points of chord be A(2cosθ,2sinθ) and B(2cosα,2sinα)
Let the midpoint be (h,k)=(cosθ+cosα,sinθ+sinα)
Given that OA and OB are perpendicular
Slope of OA is tanθ and slope of OB is tanα
So we have tanθtanα=1
tan(θα)=
tan(θα)=0
We have h2+k2=(cosθ+cosα)2+(sinθ+sinα)2
h2+k2=1+1+cos(θα)=2
Therefore the correct option is C

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