Hyperbola

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 Figure 4. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 5. Illustration by Hans & Cassidy. Courtesy of Gale Group. 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 Figure 6. Illustration by Hans & Cassidy. Courtesy of Gale Group. 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. ——