Ellipses

An ellipse forms when a right circular cone is sliced by an angled plane that doesn’t intersect its base, creating a closed curve.

Geometrically, an ellipse is the set of all points for which the sum of the distances to two fixed points—called foci—is constant.

The longest diameter is the major axis, and the shortest is the minor axis. These axes intersect at the center, and the major axis ends at the vertices.

The standard form is derived by centering the origin and placing the foci on the x-axis at -c and +c, where c is the distance from the center to each focus.

From any point on the ellipse, the total distance to both foci is the length of the major axis, expressed as a sum of two square roots. Rearranging and squaring both sides eliminates one square root. Squaring again removes all square roots, producing an equation with squared terms. Substituting the relationship between the axis lengths and focal distance, then rearranging, gives the standard form.

The larger denominator corresponds to the major axis, regardless of the ellipse’s center.

This equation describes real-world elliptical paths like planetary orbits and satellite motion.