# 7.4 Additional Properties of Hyperbolas

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 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.)

• A rational parametric representation for C is given by .
• The polar equation for C in the usual polar coordinate system is

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.)

• Let P be any point of C. The unsigned difference between the distances PF and PF' is constant, and equal to 2a.
• 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 ½ab.
• 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'.

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