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Question

A hyperbola passes through the point P(2,3) and has focii at (±2,0). Then the tangent to this hyperbola at P also passes through the point:

A
(32,23)
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B
(22,33)
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C
(3,2)
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D
(2,3)
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Solution

The correct option is B (22,33)
Coordinates of foci are (±2,0)
ae=2a2e2=4a2+b2=4b2=4a2
Thus, equation of hyperbola is x2a2y24a2=1
Given, the hyperbola passes through P(2,3),
2a234a2=1a49a+8=0a2=1,8[Ignoring a2=8 as for a2=8b2=4]a2=1

Equation of hyperbola is x21y23=1
Hence equation of tangent at P is
2x3y3=16xy3=0
From the given options only (22,33) is correct.

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