Let the domain of the function be the interval . Let the hyperbola have eccentricity and the length of the latus rectum Then is equal to: [2026]

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Solution

Q.
Let the domain of the function $f (x)$ $= \log_{3} \log_{5} \log_{7} (9 x - x^{2} - 13)$ be the interval $(m, n)$ . Let the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ have eccentricity $\frac{n}{3}$ and the length of the latus rectum $\frac{8 m}{3} .$ Then $b^{2} - a^{2}$ is equal to: [2026]

1 9

2 11

3 5

4 7

Ans.
(4)

$\log_{5} (\log_{7} (9 x - x^{2} - 13)) > 0$

$\Rightarrow 9 x - x^{2} - 13 > 7$

   $x^{2} - 9 x + 20 < 0 \Rightarrow 4 < x < 5$

   $m = 4, n = 5$

$\Rightarrow e = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \frac{5}{3} \Rightarrow \frac{b^{2}}{a^{2}} = \frac{25}{9} - 1 = \frac{16}{9}$

$\frac{b}{a} = \frac{4}{3}$

$\Rightarrow \frac{2 b^{2}}{a} = \frac{8 m}{3} \Rightarrow \frac{2 b^{2}}{a} = \frac{32}{3}$

$\Rightarrow 2 b^{2} = \frac{32}{3} \times \frac{3 b}{4} \Rightarrow b = 4, a = 3$

   $b^{2} - a^{2} = 16 - 9 = 7$

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