Calculus: Analytic Geometry and Calculus, with Vectors

538 Polar, cylindrical, and spherical coordinates

so 0 = 20 and the given angle AOP is 3q5. Thus the line OP1 trisects the given

angle. It is possible to use ruler-and-compass constructions to locate many

points on the conchoid C, but it is impossible to give explicit instructions for

producing the particular point P, needed for trisection of the given angle.

18 Let 0 be a point on a circle of diameter a, and let b be a positive number.

A set S of points P (a limacon of Pascal) is determined in the following way.

If L is a line through 0 and if Q is either the second point in which L intersects

the circle or is 0 itself if L is tangent to the circle, then S contains the two points

on L at distance b from Q. Sketch some figures and investigate these limacons.

Remark: With suitable coordinates, the equation can be put in the forms

p=acos0±b, p=acos0+b, p=b-acos0.

The fact that cos (0 + 7r) cos 0 is important.

19 Let a and b be positive constants. Let F, and F2 be two points having

polar coordinates (a,2r) and (a,0 and rectangular coordinates (-a,0) and (a,0).

The set S of points for which IF,PI JF2PI = b2 is called an oval of Cassini. Investi-

gate these ovals. Remark: If b >> a (read "if b is much greater than a"), then

S is closely approximated by a large circle. If b = a, then S is a "figure eight"

which is, in fact, a lemniscate; see Problem 6. If b <<a (read "if b is much smaller

than a"), then S consists of two small ovals that are closely approximated by

small circles.

10.2 Polar curves, tangents, and lengths As our discussion of

coordinate systems may have indicated, polar coordinates can be par-

Figure 10.21

ticularly useful in situations where dis-

tances from an origin are particularly

significant. It turns out that the polar

equation of a conic is exceptionally neat

and attractive when we put the conic in

the "standard position." As in Figure

10.21, let the focus and directrix of a conic

K having eccentricity e be placed upon a

polar coordinate system in such a way that

a focus is at the origin and the directrix is

perpendicular to the initial line and inter-

sects the extended initial line at the point having polar coordinates

(p,2r). The intrinsic equation of the conic K, which first appeared in

our work in (6.23), is then

(10.22) IPPI = eIPDI.

While it can be presumed that we know something about conics and

can proceed without the result, it is nevertheless interesting to use the

intermediate-value theorem to prove that if 7r/2 < Iq5ol 5 a, then

there is exactly one point Po with polar coordinates (pogo) for which

IFP0I = elDoPoI. In any case, we consider only values of p for which