Calculus: Analytic Geometry and Calculus, with Vectors

and hence if

(2)

If (2) holds, then

(3)

and hence either

(4)

or

(5)

6.2 Geometry of cones and conics 361

Vx2 +y2= a ± y.

x2 + y2 = a2 2ay + y2

a^2

y- 2+2a

ax'-

2

_

2a

It can be shown that if (4) or (5) holds, then (1) holds. It follows that S is the
sum (or union) of two parabolas of which one has the equation (4) and the other
has the equation (5). Each parabola has its focus at the origin, and the directrices
are the tangents to the circle that are parallel to the x axis. The parabolas
intersect the x axis where the circle does.
24 With or without the aid of results of preceding problems, let P be a given
parabola and verify the following facts which show students of mechanical draw-
ing how to locate the axis, vertex, focus, and directrix of P. The mid-points M,
and M2 of two parallel chords C, and C2 of P determine a line L, parallel to the
axis of P. The axis L of P is the perpendicular bisector of the line segment joining
points where a line perpendicular to L, intersects P. In case L, is not the axis of
P, the mid-points Mi and M2 of chords Cx and C2 perpendicular to C, and C2
determine another line Li parallel to L. Let L, and Li intersect the parabola at
P, and Pl. Then the line T, (or Ti) through P, (or P'1) parallel to C, (or Ci) is
tangent to P at P, (or Pl). Moreover T, and Ti are perpendicular, and their
intersection is on the directrix of the parabola. Finally, the line segment joining
P, and Pi is a focal chord of the parabola so this segment intersects L at the focus.
Remark: Anyone who spends a substantial part of his lifetime working with
parabolas can learn very much about them.

6.2 Geometry of cones and conics

is always a complete right circular conical surface consisting of two parts,
or nappes, as in Figure 6.21. We assume that0 < a < Ir/2
and that the lines on the cone all make the same angle a
with the axis of the cone. For the present, we simplify
our discussion by supposing that the axis of the cone is
vertical, and hence that each plane perpendicular to the
axis is horizontal. A conic (or conic section) is the set of
points in which a plane in intersects the cone. In case in
contains the vertex Y, the resulting conic is either a single
point or a single line or a pair of intersecting lines. In
case in is perpendicular to the axis of the cone and does
not contain the vertex, the conic is a circle. Our interest