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Question

If a hyperbola passes through the point P (2,3) and has foci (±2,0), then the tangent to this hyperbola at P also passes through the point

A
(32,33).
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B
(22,33).
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C
(2,3).
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D
(2,3).
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Solution

The correct option is B (22,33).
Let the equation of hyperbola be x2a2y2b2=1
ae=2a2e2=4a2+b2=4b2=4a2x2a2y24a2=1Since, (2,3) lie on hyperbola.2a234a2=182a23a2=a2(4a2)85a2=4a2a4a49a2+8=0(a28)(a21)=0a2=8,a2=1
Now equation of hyperbola is x21y23=1
equation of tangent at (2,3)is given by2x3y3=12xy3=1
which passes through the point (22,33).

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