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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)
(ae)2=4b2=4a2
Thus, equation of hyperbola is x2a2y24a2=1
Given, the hyperbola passes through P(2,3).
2a234a2=1
a2=1,8 [Ignoring a2=8 as a2<(ae)2]
a2=1
Equation of hyperbola is x21y23=1
Slope of tangent at P(2,3) is dydx(2,3)=6
Equation of tangent at P(2,3) is y3=6(x2)
Hence, the tangent passes through (22,33).

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