I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two vertices.
In the simple case of a horizontal hyperbola centred on the origin, we have the following:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
$c = \sqrt{a^2 + b^2} = a\varepsilon = a\sqrt{1 + \frac{b^2}{a^2}}$
The foci lie at $(\pm c, 0)$.
Now, if I'm not wrong about that, then this should be pretty basic algebra, but I can't see how to get from the above to an equation given a point $(x,y)$ describing the difference in distances to the foci as being $2a$. While I actually do care about the final result, how to get there is more important.
Why do I want to know this? Well, I'd like to attempt trilateration based off differences in distance rather than fixed radii.