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Question

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

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

The correct option is D (22,33)
The equation of hyperbola is x2a2y2b2=1

foci is (±2,0), so ae=2a2e2=4

b2=a2(e21)

a2+b2=4 ----- ( 1 )

Hyperbola passes through the pointv2,3

2a23b2=1 ------ ( 2 )

On solving ( 1 ) and ( 2 ), we get

a2=8 ( is rejected ) and a2=1 and b2=3

x21y23=1

Equation of tangent is 2x13y3=1

Hence, P also passes thrugh the points (22,33)

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