Calculus: Analytic Geometry and Calculus, with Vectors

10.1 Geometry of coordinate systems 537

is the polar equation of the conchoid The graph consisting of the two solid
branches nearest the line L is a conchoid for which q < p. The dotted graph
consisting of the two outer branches is a conchoid for which q > p. It can be
observed that finding the polar equation of the conchoid was no problem; the
equation came as the conchoid was being defined. Now comes the problem.
By direct use of Figure 10.193 or, alternatively, by using (4) and transformation

Figure 10.193 Figure 10.194

formulas, find the rectangular equation of the conchoid which applies to the

primed coordinate system for which the x' and y' axes are the x axis and the

line having the equation x = p. llns.:

(5) x'2y'2 = (q2 - x'2)(x' + P.

Remark: Conchoids are interesting examples of graphs of quartic equations, that

is, equations of the form f(x,y) = g(x,y), where f and g are polynomials in x and

y one of which has degree 4 and the other of which has degree not exceeding 4.

To trisect the given angle 140P of Figure 10.194 with the aid of a conchoid, let

M be the point at which the line OP intersects the line L, and let a = IOMI.

Let C be the far branch of the particular conchoid for which q = 2a. Let P1

be the point at which the horizontal line through M intersects C and let M1 be

the point at which the line OP1 intersects the line L so that 1M1P11 = 2a. To

begin our attack upon angles, let 0 be the angle ROP1 and let 8 be the angle

P1OP, so that the given angle f10P is 0 + 0. Applying the law of sines to the

triangle OMP1 gives the first of the equations

a 1MP1I a 2a cos di

sin sin 0 sin ¢ sin 8

and the second follows because 1-HP-11 2a cos 0. Thus

sin 0=2sin 4cos0=sin 241,