# Hyperbola

## Other Definitions

A hyperbola can be defined in several other ways, all of them mathematically equivalent:

- A hyperbola is a set of points P such that PF
_{1}PF_{2}= ± C, where C is a constant and F_{1}and F_{2}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 x^{2}/A^{2}- y^{2}/B^{2}= ± 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.

## Additional topics

Science EncyclopediaScience & Philosophy: *Hydrazones* to *Incompatibility*Hyperbola - Other Definitions, Features, Drawing Hyperbolas, Uses