Calculus: Analytic Geometry and Calculus, with Vectors

6.4 Hyperbolas 379

to expressthis result in terms of coordinates, we can suddenly realize that it
is almost familiar. If we put it in the form

IF1Pj + JF,PI = r,

we see anexpression of the string property of an ellipse. Therefore, S is an
ellipse having foci at the center of the circle and the given point F.

28 Sketch a figure like Figure 6.31 in which the distance from Y to Y2 is

about 10 or 20 times the distance from F to V1. Try to decide whether the

center of the ellipse is on the axis of the cone.

29 Remark: Figure 6.31 presents an interesting problem in plane geometry.

When two lines through V and a line 7r are given, we can use a ruler and compass

to construct the circles of the figure. We can then wonder whether we can give

a simple proof thatI V1F, I = 1F2721 without use of the cone and planes and spheres

that were employed in the proof in the text. Perhaps consideration of this

problem will increase our respect for the methods that were employed.

6.4 Hyperbolas Geometric properties of hyperbolas can be extracted

from Figure 6.41, which, like some

preceding ones, shows a cone hav-

ing a vertical axis. The axis lies in

the plane of the paper. The plane

x intersects the cone in ahyperbola

H of which the two points V, and

Y2 (vertices, in fact) lie in the plane

of the paper. The upper circle re-

presents a sphere, in the upper

nappe of the cone, which is tangent

to the cone at the points of a circle

which determines the plane 7r1 and

is tangent to in atF1. As we saw in

Section 6.2, irl and ir intersect in a

line L1 and, moreover, Fl is a focus

and Lx is a directrix of the hyper-

bola H. The lower circle repre-

sents a sphere, in the lower nappe of

Figure 6.41

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

determines the plane 7r2 and is tangent to it at F2. The planes 72 and N

intersect in a line L2, and the same procedure which was applied toF, and

Ll shows that F2 is another focus and L2 is another directrix of the hyper-

bola H. Thus H has two foci and two directrices.

Hyperbolas have a string property which involves the diference (not

sum) of the distances IF1PI and IF2PI from foci to points on the hyperbolas.

Let P be Vi or any other point on the upper branch of H. The line VP

lies on the cone and is tangent to the lower and upper spheres at points