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Equation of the hyperbola with focus (-3, 4),directrix 3x - 4y+ 5 = 0 and $e = \frac{5}{2}$ is

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$5 x^{2} - 24 x y + 12 y^{2} - 8 x - 6 y - 25 = 0$

$5 x^{2} - 24 x y + 12 y^{2} - 7 x - 12 y - 65 = 0$

$5 x^{2} - 24 x y + 12 y^{2} - 12 x + 8 y - 55 = 0$

$5 x^{2} - 24 x y + 12 y^{2} + 6 x - 8 y - 75 = 0$

answer is A.

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Detailed Solution

$\begin{array}{l} SP = e PM \Leftrightarrow (SP)^{2} = e^{2} {PM}^{2} \\ \Leftrightarrow 4 \{(x + 3)^{2} + (y - 4)^{2}\} = 25 {(\frac{3 x - 4 y + 5}{5})}^{2} \\ \Rightarrow 5 x^{2} - 24 x y + 12 y^{2} + 6 x - 8 y - 75 = 0 \end{array}$

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