364 Cones and conis
P(x y) know the eccentricity e, the coordinates
(x1jy1) of a locus F, and the equation
F(x,,y,) 'IX + By + C = 0 of a directrix L. While
different values of e yield conics of different
Ax+By+C=o' shapes, the schematic Figure 6.25 may beFigure 6.25 helpful. The formula JPF1 = e;PDJ is equiv-
alent to the formula IFPI2 = e2IPDI2, and
use of formulas for distances from points to points and from points to
lines (see Theorem 1.48) enables us to put this in the form
(6.251) (x - x1)2 + (y- y1)2 = A2+B2(Ax + By + C)2.
While the equation (6.251) has its virtues, we can obtain a more inform-
ative equation by choosing the x,y coordinate system in such a way that
the focus F lies on the x axis and the
directrix is perpendicular to the x axis.
With the intention of so determining x1
that the resulting equation will have its
simplest form, we suppose that the focusF
x=z1-p F(x,,O) X has coordinates (x1jO) and that the direc-
Figure 6.26 trix lies a given positive distance p to the
left of the focus (as in Figure 6.26) so
that the equation of the directrix is x = x1 - p. The intrinsic equation
IFF12=e2IPD12 then gives the coordinate equation(x - x1)2 + y2 = e2[x - (x1 - p)]2
or(6.261) (1 - e2)x2 + 2[e2(x1 - p) - x1]x + y2 = e2(x1 - p)2 - xi.We can now begin to see how the nature of the equation depends upon the
eccentricity e. In case e = 1, so that the conic is a parabola, (6.261)
reduces to(6.262) x = 2p y2 + x1 -P).
This equation has its simplest form when x1 = p/2. The simplest equa-
tion has the form x = ky2 which (except that the roles of x and y were
interchanged to simplify matters) was studied in Section 6.1.
We now face the task of simplifying (6.261) for cases in which 0 < e < I
or e > 1 and the conic is an ellipse or a hyperbola. As is easy to guess,
the greatest simplification results from so choosing x1 that the coefficient
of x is 0. Therefore, we let x1 be determined by the equivalent equations(6.27 e2 x1 - -e2p
-p
( p)-x1=0, x1=1-e22 x1-p=1- e2.