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
Answer: Option D
:
D
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
x4−r2x2+c4=0Now,CP2+CQ2+CR2+CS2
=x21+y21+x22+y22+x23+y23+x24+y24
=(∑4i=1xi)2−2∑x1x2+=(∑4i=1yi)2−2∑y1y2
=2r2+2r2=4r2 [from equation (iii)]
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:
D
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
x4−r2x2+c4=0Now,CP2+CQ2+CR2+CS2
=x21+y21+x22+y22+x23+y23+x24+y24
=(∑4i=1xi)2−2∑x1x2+=(∑4i=1yi)2−2∑y1y2
=2r2+2r2=4r2 [from equation (iii)]
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