When a hyperbola is drawn as in Figure 4, the line through the foci, F1 and F2, is the "transverse axis." V1 and V2 are the "vertices," and C the "center." The transverse axis also refers to the distance,V1V2, between the vertices.
The ratio CF1/CV1 (or CF2/CV2) 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 CV2, and whose hypotenuse CQ equals CF2. 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 = QV2.
A hyperbola is symmetric about both its transverse and its conjugate axes.
When a hyperbola is represented by the equation x2/A2 - y2/B2 = 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 x2/A2 - y2/B2 = 1 and x2/A2 - y2/B2 = -1 are called "conjugate hyperbolas.") Hyperbolas whose asymptotes are perpendicular to each other are called "rectangular" hyperbolas. The hyperbolas xy = k and x2 - y2 = ± C2 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 F1PF2 the tangent to the hyperbola at point P will bisect that angle. ——