If equation of a hyperbola whose conjugate axis is 5 and distance between it s foci is 13, is ${\mathrm{ax}}^2-{\mathrm{by}}^2=c$ where a and b are comprime natural numbers, then value of $\frac{\mathrm{ab}}{\mathrm{c}}$ is

let Equation of the hyperbola is $\frac{x^2}{a^2}-\frac{{\mathrm{y}}^2}{b^2}=1$

$\begin{array}{l}\therefore b=\frac{5}{2}\text{ and 2ae=13}\\ now b^2=a^2(e^2-1)\\ \Rightarrow b^2=a^2{e}^2-a^2\\ \Rightarrow a^2=\frac{169}{4}-\frac{25}{4}\\ \Rightarrow a^2=36\\ \therefore \text{ the hyperbola is }25{\mathrm{x}}^2-144{\mathrm{y}}^2=900\\ compare with given hyperbola\\ \Rightarrow \mathrm{a}=25,\mathrm{b}=144,\mathrm{c}=900\\ \therefore \frac{\mathrm{ab}}{\mathrm{c}}=4\end{array}$