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Question

Find the locus of the mid points of the chords of the circle x2+y2=16 which are tangents to the hyperbola 9x2−16y2=144.

A
(x2y2)2=16x2+9y2
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B
(x2+y2)2=16x2+9y2
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C
(x2+y2)2=16x29y2
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D
(x2y2)2=16x29y2
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Solution

The correct option is C (x2+y2)2=16x29y2
Given hyperbola may be written as, x216y29=1
So equation of tangent to this curve at point θ is,
xsecθ4ytanθ3=1..(1)
Now, let mid point of the chord of circle x2+y2=16 be P(h,k)
So equation of chord is given by,
hx+ky=h2+k2.......(2)
But both line (1) and (2) are same,
secθ4h=tanθ3k=1h2+k2
secθ=4hh2+k2,tanθ=3kh2+k2
Eliminating θ we get,
(4hh2+k2)2(3kh2+k2)2=1
Hence, required locus of P(h,k) is given by, 16x29y2=(x2+y2)2

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