Calculus: Analytic Geometry and Calculus, with Vectors

362 Cones and conics

in this chapter lies in conics of less simple natures. These turn out to be
parabolas and ellipses, each of which intersects only one nappe of the
cone, and hyperbolas, each of which intersects both nappes of the cone.

Figure 6.22

Without yet knowing what will be

learned, we look at Figure 6.22, which

is one of the most remarkable figures

of elementary geometry. This is a

flat nonperspective figure which must

be constructed and studied rather

carefully before it can be fully under-

stood. The vertical line in the plane

of the paper is the axis of a cone with

vertex Y and vertex angle a. The line

making the acute angle 0 with the axis

represents more than a line. It is

supposed to lie in the plane of the

paper, and it represents a plane it

which makes the angle S with the axis

of the cone and which intersects the

cone in a conic K. We can, if we wish

to do so, think of the plane 7r as being

an xy plane in which the x axis is

pointed toward our eyes, and every-

thing in this plane seems to lie on one line. The graph of each pointP on

the conic K, whether P lies on the part of the cone in front of the plane of

the paper or on the part behind the paper, is on the line. Our first

significant step is to fit a sphere into the cone, the sphere being just big

enough to be tangent to the plane ir. The circle in the figure represents

this sphere, which is tangent to the cone at the points of a circle which

lies in the horizontal plane lrl and which is tangent to the plane 7r at

the point F.

It can now be revealed that discoveries will be made; in fact we shall

show that F is a focus of the conic K. Because irl is horizontal and 7r is

not, these planes intersect in a line L which is represented by a single

point in the flat figure. We shall show that L is a directrix of the conic K.

To start learning something about the conic K, let P be a point on K

and draw the line segment PD which lies in it and is perpendicular to L

at the point D on L. The line PY lies on the cone and is tangent to the

sphere at a point A in r1. The line PF lies in 7r and is tangent to the sphere

at F. Therefore IPFI = IPA1 because the two vectors have their tails

at the same point P and are tangent to the sphere at their tips. If we

let d be the distance from P to the plane art, then d = API cos a because

the vector AP makes the angle a with vertical lines. Also d = I DPI cos 0