In this appendix to my previous post “From Dancing Planet to Kepler’s Laws“, we derive the polar form for an ellipse that has a rectangular coordinate system’s origin as one of its foci.
We start with the ellipse shown in Fig. 1. Namely,
After shifting the origin to the right by , the ellipse has the new origin as one of its foci (Fig. 2).
Since , the ellipse in is
Substituting in (2) by
Replacing by respectively, the equation becomes
Solving (3) for (see Fig. 3) gives
The first solution
The second solution is not valid since it suggests that :