Calculus: Analytic Geometry and Calculus, with Vectors

380 Cones and conics

A2 and A1. With the aid of Figure 6.41 and familiar facts about vectors

tangent to spheres at their tips, we see that

(6.42) IF2PI - IF1PI = IA2PI - IV1PI = IA2A1I.

Wherever the point P may be on the upper branch of H, the number

I72ZI is the constant sum of the slant heights of two conical segments

having their vertex at Y and their bases in the planes lrl and 72; in fact if

d is the distance between the planes 7r1 and 1r2, then IA2A1I = d/cos a,

where a is the angle at the vertex of the cone. In particular, letting

P = IV, shows that

IF21/1I - I Y1FI = 1 21I

and hence

(6.421) IF2Y2I + IY2V1I - IY1F1I = IT-42I.

In case P lies on the lower branch of H, we can reverse the roles of the

subscripts 1 and 2 to obtain the formulas

IF1PI - IF2I = IA1A21, IF1Y2I - IF2Y2I = 121,421

and

(6.422) IF1V1I + I Y2Y1I - I Tj'2I= I A122I.

Adding the formulas (6.421) and (6.422) shows that 2I72yl1 =

and hence that the first of the formulas

(6.423) ITYli = IY2YlI, IF2Y2I = IF1J

is valid. The second formula is a consequence of the first and (6.421).

All this implies the string property of the hyperbola having foci at F1 and

F2 and vertices at Yl and V2. If P is on the hyperbola, then

(6.43) IFIPI -IF2PI = I IV2I,

the plus sign being required when P is on one branch and the minus sign

being required when P is on the other branch.

Figure 6.44 shows a hyperbola and also some numerical dimensions

Figure 6.44

b 72

b^1

ae

e