# Hyperbola

## Features

When a hyperbola is drawn as in Figure 4, the line through the foci, F_{1} and F_{2}, is the "transverse axis." V_{1 }and V_{2} are the "vertices," and C the "center." The transverse axis also refers to the **distance**,V_{1}V_{2}, between the vertices.

The ratio CF_{1}/CV_{1} (or CF_{2}/CV_{2}) is the "eccentricity" and is numerically equal to the eccentricity e in the focus-directrix definition.

The lines CR and CQ are asymptotes. An asymptote is a straight line which the hyperbola approaches more and more closely as it extends farther and farther from the center. The point Q has been located so that it is the
vertex of a right triangle, one of whose legs is CV
_{2}, and whose hypotenuse CQ equals CF_{2}. The point R is similarly located.

The line ST, **perpendicular** to the transverse axis at C, is called the "conjugate axis." The conjugate axis also refers to the distance ST, where SC = CT = QV_{2}.

A hyperbola is symmetric about both its transverse and its conjugate axes.

When a hyperbola is represented by the equation x^{2}/A^{2} - y^{2}/B^{2} = 1, the x-axis is the transverse axis and the y-axis is the conjugate axis. These axes, when thought of
as distances rather than lines, have lengths 2A and 2B respectively. The foci are at

The equations of the asymptotes are y = Bx/A and y = -Bx/A. (Notice that the constant 1 in the equation above is positive. If it were -1, the y-axis would be the transverse axis and the other points would change accordingly. The asymptotes would be the same, however. In fact, the hyperbolas x^{2}/A^{2} - y^{2}/B^{2} = 1 and x^{2}/A^{2} - y^{2}/B^{2} = -1 are called "conjugate hyperbolas.") Hyperbolas whose asymptotes are perpendicular to each other are called "rectangular" hyperbolas. The hyperbolas xy = k and x^{2} - y^{2} = ± C^{2} are rectangular hyperbolas. Their eccentricity is &NA; 2. Such hyperbolas are geometrically similar, as are all hyperbolas of a given eccentricity.

If one draws the **angle** F_{1}PF_{2} the tangent to the hyperbola at point P will bisect that angle. ——

## Additional topics

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