Polar form for an ellipse offset from the origin

I recently needed a polar-form equation for an ellipse, where the center of the ellipse is offset from the origin. I couldn’t easily find such an equation, so I derived it and am posting it here. (It’s easy to find expressions for ellipses where the focus is at the origin.)

Result

The radius as a function of angle is

where $a$ is the horizontal semi-major axis, $b$ is the vertical semi-major axis, and the center of the ellipse is $(x, y)$.

In Mathematica:

rho[a_, b_, x_, y_, th_] := (a^2 y Sin[th] + b^2 x Cos[th] + 
   a b  Sqrt[ Cos[th]^2 (b^2 - y^2) + 2 x y Sin[th] Cos[th] + (a^2 - 
        x^2) Sin[th]^2])/(b^2 Cos[th]^2 + a^2 Sin[th]^2)

Limitations

This equation only works for all $\theta$ where the origin is inside the ellipse: otherwise there are some angles for which there is no radius to the ellipse, and two solutions for the radius elsewhere.

Hints for the derivation

Solve $\tan \theta = (b \sin t + y) / (a \cos t + x)$ for t. Insert the result for $t$ into \(d^2 = (a \cos(t) + x)^2 + (b \sin(t) + y)^2\).