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Question

Let chords of the circle x2+y2=a2 touch the hyperbola x2a2y2b2=1. Then their middle points lie on the curve

A
(x2y2)=a2x2b2y2
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B
(x2+y2)2=a2x2b2y2
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C
(x2y2)2=a2x2b2y2
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D
(x2+y2)=a2x2b2y2
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Solution

The correct option is B (x2+y2)2=a2x2b2y2
The equation of the chord of the circle x2+y2=a2 whose middle point is (α,β) is
xα+yβ=α2+β2y=αβx+α2+β2β
Now the above equation is the tangent to the hyperbola
using the tangency condition
m=αβ,c2=a2m2b2(α2+β2β)2=a2α2β2b2
(α2+β2)2=a2α2b2β2
Hence, the locus of the middle point is
(x2+y2)2=a2x2b2y2

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