Let the lengths of the transverse and conjugate axes of a hyperbo… - Vidyadip

Question

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b , respectively, and one focus and the corresponding directrix of this hyperbola be $(-5,0)$ and $5 x+9=0$, respectively. If the product of the focal distances of a point $(\alpha, 2 \sqrt{5})$ on the hyperbola is $p$, then $4 p$ is equal to __________ .

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Answer

189
$\mathrm{PF}_{1} \cdot \mathrm{PF}_{2}=(\mathrm{e} \alpha-\mathrm{a})(\mathrm{e} \alpha+\mathrm{a})$
$\mathrm{P}=\mathrm{e}^{2} \alpha^{2}-\mathrm{a}^{2}=\frac{25}{9} \cdot 9 \cdot \frac{9}{4}-9=\frac{189}{4}$
$\left.\begin{array}{c}a e=5 \\ \frac{a}{e}=9 / 5\end{array}\right\} \begin{gathered}a=3 \\ e=5 / 3 \\ b=4\end{gathered}$
$\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$
$\frac{\alpha^{2}}{9}-\frac{4(5)}{16}=1 \Rightarrow \alpha^{2}=9\left(\frac{36}{16}\right)$

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