Find hyperbolas transverse and conjugate axes
The product of distances from focis to any tangent line is the square of conjugate semiaxis.
The distances from $F_1$ and $F_2$ to the tangent line are respectfully $15/5$ and $48/5$, so $b=12/\sqrt{5}$.
The distance from $F_1$ to $F_2$ is equal to $2c$, then $4c^2=13^2$.
Then $a= \sqrt{c^2-b^2} = \sqrt{269}/(2\sqrt5)$.