** Next:** 7.5 Additional Properties of Parabolas
**Up:** 7 Conics
** Previous:** 7.3 Additional Properties of Ellipses

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
**parametric representation**for*C*is (*a*sec ,*b*tan ).A different parametric representation, which gives one branch only, is

(

*a*cosh ,*b*sinh ):The

**area**of the shaded sector above isThe

**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.) - A
**rational parametric representation**for*C*is given by . - The
**polar equation**for*C*in the usual polar coordinate system isWith 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.) - Let
*P*be any point of*C*. The unsigned**difference between the distances***PF*and*PF'*is constant, and equal to 2*a*. - Let
*P*be any point of*C*. Then the rays*PF*and*PF'*make the same angle with the tangent to*C*at*P*. Thus any light ray originating at*F*and reflected in the hyperbola will appear to emanate from*F'*. - Let
*T*be any line tangent to*C*. The product of the distances from*F*and*F'*to*T*is constant, and equals*b*. - Let
*P*be any point of*C*. The area of the parallelogram formed by the asymptotes and the parallels to the asymptotes going through*P*is constant, and equals ½*ab*. - Let
*L*be any line in the plane. If*L*intersects*C*at*P*and*P'*, and intersects the asymptotes at*Q*and*Q'*, the distances*PQ*and*P'Q'*are the same. If*L*is tangent to*C*we have*P*=*P'*, so the point of tangency bisects the segment*QQ'*.

*Silvio Levy
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.