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Question

A rectangular hyperbola whose centre is C, is cut by any circle of radius r in four points P, Q, R and S. Then, CP2+CQ2+CR2+CS2 is equal to


A

r2

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B

2r2

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C

3r2

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D

4r2

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Solution

The correct option is D

4r2


Let equation of the rectangular hyperbola be xy=c2 and equation of circle be x2+y2=r2
Put y=c2x in equation (ii), we get x2+c4x2r2
x4r2x2+c4=0Now,CP2+CQ2+CR2+CS2
=x21+y21+x22+y22+x23+y23+x24+y24
=(4i=1xi)22x1x2+=(4i=1yi)22y1y2
=2r2+2r2=4r2 [from equation (iii)]


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