What Is an Ellipse?

How axes and foci fit together

Analytic geometry also treats the ellipse as a conic section formed when a plane cuts a cone at an angle that produces a closed curve but not a circle. In coordinates, its equation shows how far the graph reaches horizontally and vertically from its center.

This page keeps the foci, axes, and curve together so the ellipse is read as a structured conic rather than as a loosely stretched circle shape.

An ellipse with a longer horizontal axis is not described by one ordinary radius. Its semi-major and semi-minor axes measure different directions, while the two foci sit along the longer axis. The fixed-sum distance rule explains why the curve closes instead of opening like a parabola.

• An ellipse is the set of points whose distances from two foci have a constant sum.
• An ellipse has a center and two axes: the major axis is longer, and the minor axis is shorter.
• If the two axes are equal in length, the ellipse becomes a circle.
• The foci lie on the major axis, and the sum-of-distances property stays constant for every point on the curve.

Where ellipses help

• Use ellipses in coordinate-conic problems involving foci, axes, and standard-form equations.
• Use them in modelling orbital paths, reflective properties, and design shapes.
• Use ellipse graphs when comparing conics that are closed curves but not circles.

Ellipse checks to make

• Do not confuse the center with the foci; the foci are separate points on the major axis.
• Do not use the sum-of-distances rule as if it were a difference-of-distances rule; that belongs to hyperbolas.
• Do not assume every oval-looking graph is an ellipse unless the coordinate structure matches.