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Question

If tangents OQ and OR are drawn to variable circles having radius r and the centre lying on the rectangular hyperbola xy=1, then locus of circumcentre of triangle OQR is (O being the origin).

A
xy=4
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B
xy=14
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C
xy=1
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D
xy=4
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Solution

The correct option is B xy=14
Let S be any point on the rectangular hyperbola (t,1t)
Hence eqaution of the circle with radius r is
(xt)2+(y1/t)2=r2
Equation of chord of contact from (0,0) will be
xtyt+t2+1t2r2=0
Family of circles equation passing through P,Q is
(xt)2+(y1/t)2r2+k(xtyt+t2+1t2r2)=0
If it passes through (0,0)
k=1
Hence circumcircle equation for triangle OQR will be
(xt)2+(y1/t)2r2+xt+ytt21t2+r2=0x2+y2xtyt=0
whose centre is (t/2,1/2t)(h,k)
required locus equation will be
xy=14

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