Figure 8. 



Given the condition o^ — 02 = 2a 

 By the law of cosines applied to triangles MQQi IVlQQj 



ff/ = r^ + c^ - 2rc cos a, a^ = r^ + c^ + 2rc cos a 

 whence a^ + ct/ = 2(r' + c') , a,' ff^'' = (r'^ + c')' - 4r^ c^ cos^a 

 Now by squaring both sides of o-^- a^,' 2a obtain 



a-^ - 2ai CTj "*" '^■1 ^ 4a whence 



(a,' +a,'-4a')^ = 4a. V,' 

 With the values of Oi + ffj^' o^ 02 from (51) placed in (52) obtain 



[2(r^ + c') - 4aT = 4[(r^ + c^)^ - 4r^c^ cos^ a] . 

 Expanding (53) find 



r^ c^ cos^ a - a^ r^ - a^ c^ + a" = 



or a'(c' - a') 



r' = 



22 2 



c cos a - a 



To transform to rectangular equation we have x = r cos a, y = r sin a, or r = x + y , 



tan a = _ , cos^ a = x^/(x^ + y^) and these values of r^ and cos^a placed in (54) give 



2 2 

 a y 

 2 _ ^ , 2 

 X + a 



„2 2 



c -a 

 as corresponding rectangular equation. 



(51) 



(52) 

 (53) 



(54) 



(55) 



35 



