Ellipses

An ellipse is the set of points such the sum of the distances between a point and two fixed points, the foci, is constant and greater than the distance between the foci.

A standard ellipse centered at the origin with length $2a$ along the $x$-axis, and $2b$ along the $y$-axis, is

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 .\]

This ellipse will have foci at $(\pm \sqrt{a^2 - b^2}, 0)$, since the sum of the distances between the foci and a point of the ellipse on the $x$-axis is $2a$.

Circles

A circle is an ellipse where the axes have the same length.