Calculus: Analytic Geometry and Calculus, with Vectors

6.1 Parabolas 355

point must therefore satisfy the equation of the parabola, so? = k(2 ?) 2 and

? = 1/4k. The coordinates of F are therefore (0, 1/4k), and the equation

of the directrix is y = - 1/4k. The square of Figure 6 11 havinga vertex

x

Directrix D

Figure 6.11

atF and two vertices on the directrix is called a focal square of the parabola.
Another focal square lies to the left of the one in the figure. A figure
which shows a parabola together with its focus and directrix is imperfect
unless the parabola contains a vertex of each focal square.
The y axis, being an axis of symmetry and the only one, is called the
axis of the parabola. The point in which the parabola intersects its axis
is called the vertex of the parabola. More definitions will appear in the
problems. While parabolas have important applications in which foci
(plural of focus) and directrices (plural of directrix) never appear, most of
the problems involve situations in which they do appear.
Since preliminary ideas can be very valuable, we look briefly at Figure
6.12. The figure gives six views of the intersection of a cone and a plane

VV VV

Figure 6.12

The cone is a right circular conical surface a part of which resembles a

conical paper cup or ice-cream cone. The vertex V and the axis of the

cone are in the plane of the paper. The intersecting plane is parallel to