Question
Let the domain of the function $f(x) = \log_{3}\log_{5}\log_{7}$ ( $9x - x^{2} - 13$ ) 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 $\frac{8m}{3}$. Then $b^{2} - a^{2}$ is equal to :
Options
Solution
$\log_{5}\left( \log_{7}\left( 9x - x^{2} - 13 \right) \right) > 0$
$$\begin{matrix} \Rightarrow & 9x - x^{2} - 13 > 7 \\ & x^{2} - 9x + 20 < 0 \Rightarrow 4 < x < 5 \\ & m = 4,n = 5 \\ \Rightarrow & e = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \frac{5}{3} \Rightarrow \frac{b^{2}}{a^{2}} = \frac{25}{9} - 1 = \frac{16}{9} \\ & \frac{b}{a} = \frac{4}{3} \\ \Rightarrow & \frac{2b^{2}}{a} = \frac{8m}{3} \Rightarrow \frac{2b^{2}}{a} = \frac{32}{3} \\ \Rightarrow & 2b^{2} = \frac{32}{3} \times \frac{3b}{4} \Rightarrow b = 4,a = 3 \\ & b^{2} - a^{2} = 16 - 9 = 7 \end{matrix}$$
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