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 x^2/(a^2)-y^2/(b^2)=1 have eccentricity…

Let the domain of the function $\displaystyle f(x)=\log _3 \log _5 \log _7\left(9 x-x^2-13\right)$ be the interval (m, n). Let the hyperbola $\displaystyle \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ have eccentricity $\displaystyle \frac{\mathrm{n}}{3}$ and the length of the latus rectum $\displaystyle \frac{8 \mathrm{~m}}{3}$. Then $\displaystyle \mathrm{b}^2-\mathrm{a}^2$ is equal to :