POLAR SOLUTION 

 The following procedure involves tables of the trigonometric functions but no root extraction. 

 First express the equations of (1) and (2) in polar form both referred to the common focus F(c,o), 

 and the corresponding rectangular coordinates in terms of the polar parameters. Find for equation 

 (1) 



(c>a) (see equation (3) PLANE, page 37 with R = r , /3 = 0) 



± a - c cos 

 X = c + r cos d, y = T sin 6 (7) 



and for equation (2) 



(d^ - b^) [d cos {6 ~ a)± h] 



(d>b) 



A' cos' id - a) - h' 

 X = c + rL cos ^, y = ri sin (8) 



Since (7) and (8) express the two hyperbolas in polar form with respect to the same pole 

 F(c,o), a common focus of the two loci, it is clear (see Figure 24) that at a point of intersection 

 P'(x,y) the two values r and rt are equal to a common value r'for a common value of and 



the distances to P'from F'and F"are then given by r, = r'+ 2a, rn = r'+ 2b. 



Equating the values of r , ri from (7) and (8) one obtains 



c' -a' d'- h' 

 r'_ = 



± a - c cos 9 d cos {9 - a) ±b (9) 



and since c, d, a are constants, (9) is a relation between the parameters a, b, and 6. That is 



given any two of the three the third may be found from (9). 



Consider a and b given. First write (9) in the form 



dcos((9-a)+b d^-b'' ^ , 



— = = K., whence 



±a - c cos 9 c^ - a^ 



(d cos a + cK) cos 9 + (d sin a) sin 9 = ± aK ±h. (10) 



The solution of the trigonometric equation (10) is 

 9.=P^y. 



tan /S = (d sin a)/(d cos a + cK) (i = 1,2,3,4) 



cos y- = (+ aK + b) sin /3 /d sin a. (11) 



From (11) it is seen that in general there will be four angles (yj), and thus four values 



104 



