Let one focus of the hyperbola be at and the corresponding directrix be . If e and respectively are the eccentricity and the length of the latus rectum of H, then is equal to : [2025]

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Solution

Q.
Let one focus of the hyperbola $H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ be at $(\sqrt{10}, 0)$ and the corresponding directrix be $x = \frac{9}{\sqrt{10}}$ . If e and $l$ respectively are the eccentricity and the length of the latus rectum of H, then $9 (e^{2} + l)$ is equal to : [2025]

1 16

2 15

3 12

4 14

Ans.
(1)

We have, $a e = \sqrt{10} and \frac{a}{e} = \frac{9}{\sqrt{10}}$

$\Rightarrow a^{2} = 9 and e = \frac{\sqrt{10}}{3}$

Now, ${(a e)}^{2} = a^{2} + b^{2} \Rightarrow 10 = 9 + b^{2} \Rightarrow b^{2} = 1$

and $l = \frac{2 b^{2}}{a} = \frac{2 (1)}{3} = \frac{2}{3}$

$∴ 9 (e^{2} + l) = 9 (\frac{10}{9} + \frac{2}{3}) = 10 + 6 = 16$

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