6
Cones and conics
6.1 Parabolas Before plunging into the general aspects of this
chapter, we obtain more information about the parabolas that were
introduced in Section 1.4. Being realistic, we face some facts. We
remember that, for some strange reason, the graph of y = kx2 is, when
k > 0, a parabola, but details involving the focus and directrix of this
parabola may have been quite thoroughly forgotten. We try to recall,
and henceforth remember, that the parabola has a focus F and a directrix
as in Figure 6.11 and that the parabola is the set of points P(x,y) for
which IFPI _ 1DPI. We have forgotten how the coordinates of F and the
equation of the directrix are related to k, and we may forget again, so we
should know how to discover the facts. To put a little variety into our
lives, we use the symbol "?" to represent the unknown distance from the
origin to F and from the origin to the directrix. Now we make the key
observation. The points on the horizontal line through F all lie at dis-
tance (2?) from the directrix. Hence the point (2?,?) which lies (2?)
units to the right of F must lie on the parabola. The coordinates of this
3"