I don't get the idea how the equation for this family of curve is $\displaystyle y^2 + (x - a \coth v)^2 = \frac{a^2}{\sinh ^2v}$ from this article on Wikipedia. Suppose, the equation is $\displaystyle y^2 + (x-h)^2 = r^2$, how do I parametrize $h$ and $r$? What points should I check to get in this form?
The circle seems to have radius $0$ when it passes through $(\pm a, 0)$ and infinite radius at $(0, 0)$. What would be equation of this family of curve in Cartesian coordinate?