Calculus: Analytic Geometry and Calculus, with Vectors

6.2 Geometry of cones and conics 363

because the vector DP makes the angle S with vertical lines. Equating

the two expressionsfor d gives the formula

I API cos a = DPI cos Q.

This and the fact thatIPFI = IPDI give the fundamental formula

(6.23)

IPFI _cos P IPDI

= eIPDI

cos a

where the constant e defined by the formula

(6.231)

e = cos 0,

cos a

is called the eccentricity of the conic. The point F and the line L are

respectively a focus and a directrix of the conic. It is clear from Figure

6.22 that, whenever a and S are given acute angles, we can make the dis-

tance from F to L have any positive value p we please by taking it at the

appropriate distance from V. The equation (6.23) is called an intrinsic

equation of the conic, that is, an equation that depends only upon the

conic itself and not upon the coordinates of a particular "external"

coordinate system.

In case 0 = a, the conic is called a parabola. In this case, (6.231)

shows that e = 1 and that the formula (6.23) reduces to the simpler

formula IFFI = IPDI. Thus we have, as was promised in Section 1.4,

proved that a parabola is the set of points P (in a plane) equidistant from

a fixed point (the focus F) and a fixed line (the directrix L).

In case a < $ < 7r/2, the conic is called an ellipse. In this case

(6.24) IPFI = eIPDI

where the eccentricity is a constant e for which 0 < e < 1. As we can see

from Figure 6.22, an ellipse is an oval (or oval curve) that lies entirely

on one nappe of the cone. Ellipses will be studied in greater detail in

Section 6.3.

In case 0 < R < a, the conic is called a hyperbola. In this case

(6.241) IPFI = eIPDI,

where the eccentricity is a constant e for which e > 1. As we can see from

Figure 6.22, a hyperbola consists of two branches (or parts) one of which

is contained in each nappe of the cone. Hyperbolas will be studied in

greater detail in Section 6.4.

The above information enables us to find the equation of a nontrivial

conic K (parabola, ellipse, or hyperbola) which lies in an xy plane when we