hyperbola

A hyperbola is an open curve in the plane with two distinct branches, defined as the set of all points the difference of whose distances from two fixed points is a constant.

Algebraically, a hyperbola in the Cartesian plane with horizontal axis is a relation defined by an equation of the form

\[\displaystyle\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=r^2\]

where the point \((h,k)\) is the center of the hyperbola. The major axis of the hyperbola is vertical if the \(y\) term is positive and the \(x\) term is negative.

Figure 1: A hyperbola in the Cartesian plane.

A hyperbola also has directices and important reflection properties. See the article on conics for an exposition.