The book asked me to find the constant differences for a hyperboka with the given foci and point on the hyperbola.

With:

f(1): (-13,0)
f(2): (13,0)
P: (5,0)


The answer I got was 15, is that right?

I am sorry, but I do not know what you mean by a 'constant difference'. But, the equation of the hyperbola would be:

\frac{x^{2}}{5^{2}}-\frac{y^{2}}{12^{2}}=1

Does that help? I do not know where your 15 comes from.

Can you please elaborate?