Find Transverse axis of a Hyperbola

Question

A hyperbola touches y axis and has its centre at (5/2,20)and one of the focii at (10,24) respectively , find length of the transverse axis. well I tried to make set of equations with the general properties of a Hyperbola. But couldn't solve them to get the value of 'a' to get '2a' as length of Transverse axis

Answer 1

Hints:

1. You have the center, therefore you actually have two foci (and two tangents).

2. Mirroring a focus across a tangent gives a point at distance $2a$ from the other focus.