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Let C be the hyperbola with equation x/a-y/b=1, and let F,F'=(±,0) be its foci (see Figure 7.1.3 ). The conjugate hyperbola of C is the hyperbola C' with equation -x/a+y/b=1. It has the same asymptotes as C, the same axes (transverse and conjugate axes being interchanged), and its eccentricity e' is related to that of C by 1/e'+1/e=1.
A different parametric representation, which gives one branch only, is
(a cosh , b sinh ):
The area of the shaded sector above is
The length of the arc from (a,0) to the point
(a cosh , b sinh )
is given by the elliptic integral
where e is the eccentricity, , and x=a cosh . (See the Standard Math Tables and Formulas for elliptic integrals.)
With respect to a system with origin at a focus the equation is
where l=b/a is half the latus rectum. (Use the - sign for the focus with positive x-coordinate and the + sign for the other.)
Wed Oct 4 16:41:25 PDT 1995
This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.