Lesson 3: Ellipses

Property

An ellipse is the set of all points in a plane such that the sum of the distances from any point on the ellipse to two fixed points called foci is constant. This constant sum property creates the characteristic oval shape of an ellipse.

Examples

Property

An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse. A line drawn through the foci intersects the ellipse in two points, called a vertex of the ellipse. The segment connecting the vertices is called the major axis. The midpoint of the segment is called the center of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.

Examples

• An ellipse has foci at $(-4, 0)$ and $(4, 0)$. For any point $(x, y)$ on the ellipse, the sum of the distances from that point to each focus is a constant value.
• If an ellipse has vertices at $(6, 0)$ and $(-6, 0)$, its major axis has a length of 12 and its center is at the origin $(0, 0)$.
• If the major axis connects $(0, 7)$ and $(0, -7)$ and the minor axis connects $(-3, 0)$ and $(3, 0)$, the center of the ellipse is their intersection point, $(0, 0)$.

Explanation

Think of an ellipse as a stretched circle with two center points, called foci. The total distance from any point on the curve to both foci is always the same. This constant sum is what creates the unique oval shape.

Property

The standard form of the equation of an ellipse with center $(0, 0)$ is

The $x$-intercepts are $(a, 0)$ and $(-a, 0)$.
The $y$-intercepts are $(0, b)$ and $(0, -b)$.
When $a > b$, the major axis is horizontal. When $b > a$, the major axis is vertical.