# How can I interpolate between two points on an ellipse given only the two points in polar coordinates and the ratio of a and b?

If you have two points in polar coordinates, $p_1$ and $p_2$, and you have a ratio $k = a/b$ ( where a and b are parameters of an equation for an ellipse ), how can you find the radius for a point $p$ with angle $\theta$ between $p_1$ and $p_2$ along an ellipse parametrized by $a$ and $b$.

i.e. I have

\begin{align} r &=\frac{ab}{\sqrt{ (bcos\theta)^2 + (asin\theta)^2 } }\\ p_1 &= ( r_1, \theta_1 )\\ p_2 &= ( r_2, \theta_2 )\\ k &= \frac{a}{b}\\ p &= ( r, \theta ) \end{align}

,where only $a,b$ and $r$ are unknowns, and I would like to solve for $r$.