Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE HYPERBOLA

137

Thus the locus is seen to be a hyperbola with its center, 0',
at the point (- m 1 0) the semi-axes being given by (4).
c2 -- I
The value of c is given by the equation c2 = a2 + b2. Hence
E2
(5) c =^
The eccentricity, e =c/a, is seen to be precisely c:
e =,
and thus the given constant, c, turns out to be the eccentricity of the
hyperbola.
Finally, F is one of the foci. For, the distance from O' to F
is
0'0+ OF= +m==
Ie2 -- 1 2 2 1
and this, by (5), is precisely c.
The line D is called a directrix of the hyperbola. Its distance from the center is
nm em 1 a
0~O _2... =.
e2 -_ 1?-1 e e
The Directrices. There is a second directrix, namely, the
line D' symmetric to D in the conjugate axis. It is clear
from the symmetry of the figure that what is true of the hyperbola with respect to the focus F and the corresponding
directrix D is equally true with respect to the focus F' and
the directrix D'. Accordingly, the hyperbola is the locus of a
point whose distance from a focus bears to its distance from
the corresponding directrix a fixed ratio, the eccentricity.
The equations of the directrices of the hyperbola,
x2 y2 1
a2 b2
a a
are x=- and X

e

e