if $S(5,12)$ and $S'(24,7)$ are the Foci of hyperbola passing through $(0,0)$, find its Equation.
My Try: By the definition of Hyperbola, if $P$ is any point on it and if $S$ and $S'$ are the foci, Then
$$|PS-PS'|=2a$$ Now using $P$ as $(0,0)$ we get $$|13-25|=2a$$ $\implies$ $a=6$
Now if $P(x,y)$ is any point we have
$$PS^2+PS'^2-2PSPS'=4a^2=144$$ using distance formula and rearranging the terms we get
$$(x^2+y^2-29x-19y+325)^2=(x^2+y^2-10x-24y+169)(x^2+y^2-48x-14y+625)$$
if we simplify above we get equation of the form $$ax^2+2hxy+by^2+2gx+2fy=0$$
But i feel this algebra as lengthy is there any other method to find the equation?