## Definition and vocabulary

### Ellipse

An ellipse is the set of points around two points, called focal points, such that the sum of the distances from any point on the ellipse to the two focal points is constant.

The following graphic contains an ellipse and the two focal points are labeled

A circle is an ellipse whose two focal points are the same point, namely the center of the circle.

### Center of an ellipse

The midpoint of the two focal points of an ellipse is the center of the ellipse.

The following graphic contains an ellipse. The two focal points are labeled A and B and the center of the ellipse is labeled C.

## Properties of ellipses

### Vertices and major axis of an ellipse

The line through the two focal points of an ellipse intersects the ellipse at two points. These two points are referred to as the vertices of the ellipse. The line segment joining the vertices of an ellipse is the major axis of the ellipse.

The following figure contains an ellipse. Its vertices are labeled E and F and the line segment joining these two points is the major axis of the ellipse.

### Co-vertices and minor axis

The line passing through the center of an ellipse, which is perpendicular to the major axis, intersects the ellipse at two points. These two points are the co-vertices of the ellipse and the line segment joining the two co-vertices is the minor axis of the ellipse.

The following figure contains an ellipse. The co-vertices are labeled G and H and the line segment joining these two points is the minor axis of the ellipse.

### Area of an ellipse

Suppose we are given an ellipse as in the following graphic:

The area of the ellipse is:

** \pi r_1 r_2 **

The area of the ellipse is:

\pi\left(3\right)\left(5\right)=15\pi

## Equation of an ellipse

### Equation of an ellipse

The set of points whose coordinates \left(x,y\right) satisfy the equation:

** \dfrac{\left(x-a\right)^2}{r_1^2}+\dfrac{\left(y-b\right)^2}{r_2^2}=1 **

is an ellipse centered at \left(a,b\right). The major and minor axes are parallel with the coordinate axes, the horizontal axis has length 2r_1, and the vertical axis has length 2r_2.

Consider the following equation:

\dfrac{\left(x-5\right)^2}{9}+\dfrac{\left(y-4\right)^2}{4}=1

The set of points whose coordinates \left(x,y\right) are solutions to the above equation is an ellipse centered at \left(5{,}4\right). The major axis is horizontal and with a length of 2\sqrt{9}=2\left(3\right)=6 and the minor axis is vertical with a length of 2\sqrt{4}=2\left(2\right)=4.

When r_1=r_2, the ellipse is a circle.