No I can't do that. That's just a circle with radius = 1. What I want to do is have the equation of an ellipse with the center of the coordinate system centered at one of the foci. From wiki, I know that the answer should be:

with

I am actually using this in a physics problem showing that the orbital velocity at the point which the minor axis meets the elliptical orbit is the geometrical mean of the velocities at the ends of the major axis. With this identity:

my proof works very well, but I don't want to just pull an equation off wiki and rest my proof on that. I would like to derive that equation by doing a shift of coordinates.