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Question

If two points P and Q on the hyperbola x2a2−y2b2=1, with centre C be such that CP is perpendicular CQ, a<b then 1CP2+1CQ2 is equal to

A
1a2+1b2
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B
1a2+1b2
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C
1a21b2
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D
1a21b2
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Solution

The correct option is D 1a21b2
Since the line CP passes through the origin, i.e., centre, let its equation be y=mx. The line CP meets the hyperbola

x2a2y2b2=1 in P whose abscissa is given by

x2a2m2x2b2=1 or x2=a2b2b2a2m2

y2=m2x2=a2b2m2b2a2m2
CP2=x2+y2

=a2b2+a2b2m2b2a2m2=a2b2(1+m2)

Since CQCP

Replace m by 1m, we get

CQ2=a2b2(1+1m2)b2c2m2=a2b2(m2+1)b2m2a2

1CP2+1CQ2=b2a2m2+b2m2a2a2b2(1+m2)=b2a2a2b2

=1a21b2

Hence, option D.

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