A hyperbola can be defined in several other ways, all of them mathematically equivalent:
- A hyperbola is a set of points P such that PF1 PF2 = ± C, where C is a constant and F1 and F2 are fixed points called the "foci" (see Figure 2). That is, a hyperbola is the set of points the difference of whose distances from two fixed points is constant. The positive value of ± C gives one branch of the hyperbola; the negative value, the other branch.
- A hyperbola is a set of points whose distances from a fixed point (the "focus") and a fixed line (the "directrix") are in a constant ratio (the "eccentricity"). That is, PF/PD = e (see Figure 3). For this set of points to be a hyperbola, e has to be greater than 1. This definition gives only one branch of the hyperbola.
- A hyperbola is a set of points (x,y) on a Cartesian coordinate plane satisfying an equation of the form x2/A2 - y2/B2 = ± 1. The equation xy = k also represents a hyperbola, but of eccentricity not equal to 2. Other second-degree equations can represent hyperbolas, but these two forms are the simplest. When the positive value in ± 1 is used, the hyperbola opens to the left and right. When the negative value is used, the hyperbola opens up and down.