Ellipses are conic sections formed when a plane intersects a cone in an inclined way. The main characteristic of ellipses is that all the points on their curve have a sum of distances from two fixed points that is equal to a constant. The two fixed points are called the foci of the ellipse.

Here, we will look at a more detailed definition of ellipses along with an illustration to visualize it. Then, we will learn about the most important characteristics of these conic sections.

PRECALCULUS
definition of an ellipse with foci

Relevant for

Learning about the fundamental characteristics of ellipses.

See characteristics

PRECALCULUS
definition of an ellipse with foci

Relevant for

Learning about the fundamental characteristics of ellipses.

See characteristics

Definition of an ellipse

Ellipses are defined as the set of all points in the Cartesian plane, which have two distances from two fixed points that add up to always obtain a constant. These two points are known as the foci of the ellipse and serve to define it.

definition of an ellipse with foci

Additionally, we can define ellipses as conic sections that are obtained by the intersection of a plane with a cone when the plane is inclined at an angle with respect to the base of the cone.

Ellipses have two axes of symmetry, the major axis, and the minor axis. The major axis is the longest diameter of the ellipse (usually denoted by a). The major axis extends from one end of the ellipse to the other at the widest part and passes through the center.

On the other hand, the minor axis is the shortest diameter (denoted by b). The minor axis crosses through the center at the narrowest part of the ellipse.


Main characteristics of an ellipse

The main characteristics of an ellipse are:

  • The ellipse has two focal points, called the foci.
  • The eccentricity of the ellipse is between 0 and 1.
  • The total sum of each distance from a point on the ellipse to the two foci is constant.
  • Ellipses have a major axis and a minor axis.
  • The intersection of the major axis and the minor axis is the center of the ellipse.
  • A circle is a special case of an ellipse, which has both foci in the center.

Equation of an ellipse

Depending on the location of the center of the ellipse, we can have two variations of its equation. When the center is located at the origin, that is, at the point (0, 0), the equation of the ellipse is:

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

where a represents the length of the semi-major axis and b represents the length of the semi-minor axis.

If the center is not located at the origin, the equation of the ellipse is:

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

where, (h, k) is the center of the ellipse.


See also

Interested in learning more about ellipses? Take a look at these pages:

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