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The point of intersection of two tangents to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,$ the product of whose slopes is $c^{2},$ lies on the curve,

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$y^{2} + b^{2} = c^{2} (x^{2} - a^{2})$

$y^{2} - a^{2} = c^{2} (x^{2} + a^{2})$

$y^{2} - b^{2} = c^{2} (x^{2} + a^{2})$

$x^{2} + b^{2} = c^{2} (x^{2} - a^{2})$

answer is B.

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Detailed Solution

$y = m x + \sqrt{a^{2} m^{2} - b^{2}}$

$y - m x - \sqrt{a^{2} m^{2} - b^{2}} = 0$

$y^{2} + m^{2} x^{2} - 2 m x y = a^{2} m^{2} - b^{2}$

$k^{2} + m^{2} h^{2} - 2 m h k = a^{2} m^{2} - b^{2}$

$m^{2} (h^{2} - a^{2}) - 2 m h k + b^{2} + k^{2} = 0$

$\frac{b^{2} + k^{2}}{h^{2} - a^{2}} = c^{2}$

$b^{2} + y^{2} = c^{2} (x^{2} - a^{2})$

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