Ellipse

An ellipse consists of all the points in a plane, the sum of whose distances from two fixed points (the foci) in the plane is constant.
In the figure, the foci are labeled. The sum of the lengths of the green and blue line segments equals the length of the red line segment.

Denote the sum of the distances by 2a and suppose that the foci are located at the two points (-c,0) and (c,0). Set b=\sqrt{a^2-c^2}. (Why is a^2\ge c^2?)
A point (x,y) lies on the ellipse if and only if

    \[\sqrt{(x-c)^2 +(y-0)^2} + \sqrt{(x-(-c))^2 +(y-0)^2} = 2a\]

Direct calculation can be used to put this equation into the form

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

The four points (a,0), (-a,0), (0,b), and (0,-b) all lie on the ellipse because

    \[{(\pm a)^2 \over a^2} + {0^2 \over b^2} = 1\]

    \[{0^2 \over a^2} + {(\pm b)^2 \over b^2} = 1\]

The points (a,0) and (-a,0) are called vertices of the ellipse. The line from (-a,0) to (a,0) is called the major axis and the line from (0,-b) to (0,b) is called the minor axis.

The standard equation for an ellipse with center (h,k) and semi-axes a and b is given by

    \[\frac{\left( x-h\right) ^{2}}{a^{2}}+\frac{\left( y-k\right) ^{2}}{b^{2}}=1\]

A parametric representation for this ellipse is given by

    \[( h+a\cos t,\ k+b\sin t)\]