Critique of rectangular coordinate equation
The end of the proof and derivation illustrate what Sellers means by the "algebraic side of ellipses." Having found an equation for ellipses, we can manipulate it algebraically to get it into other forms. And we might ask how changing the ellipse in other ways, such as by rotation or translation, would affect the equation.
We might also ask, "What if not rectangular coordinates?" What would the equation be if we used polar coordinates? What would the parametric equations of an ellipse be?
The rectangular coordinate equation for an ellipse looks a lot like that of a unit circle:
