68 AVALYTIf GEOMETRY [Cn. III. 



EXERCISES 



1. Verify Art. 41 by first finding the coordinates of th- poii 

 int.Ts.M-tioii of the loci of equations (1) and (2), and then substituting 

 these coordinates in equation (3). 



2. Kind the equation of a curve that passes through all the points in 

 which the following pairs of curves intersect: 



y = 2 cos x. 



3. Find (lie equation of a curve through all the points common to the 

 following pairs of curves : 



NOTE. It is to be observed that the method given in Art. 39, for find- 

 ing the point of intersection of two curves, is an application of tip- 

 theorem of Art. 41. For the process of solving two simultaneous equa- 

 tions, at least one of which involves two variables, consists in combining 

 them in such a way as to obtain two simple equations, each involving 

 only one variable. Now each of these simple equations represents an 

 elementary locus, one or more straight lines parallel to the axes, if the 

 coordinates are Cartesian ; circles about the pole, or straight lines through 

 the pole, if the coordinates are polar, and these elementary loci deter- 

 mine, i.e., pass through, the points of intersection of the original loci. 

 To determine the points of intersection, then, of two loci, the original 

 loci are replaced by simpler ones passing through the same common 

 points. E.g., the points of intersection of the loci of Art. 39, 



2y-z=0 . . . (1), and y* = x, ... (2) 

 are given by the equations 



( v *-z)-(2y-z)=0 and [(2y) - a*} - 4 (y* - x) = 0, 

 that is, by y* - 2 y = 0, and x 9 - 4 x = 0, 



which may be written 



y(y-2) = . . . (3), zrz-4) = 0. ... (4) 



But the locus of equation (3) is a pair of straight lines parallel to the 

 z-axis, and the locus of equation (4) is a pair of straight linen parallel 

 to the y-axis ; and these loci have the same points of intersection as the 

 loci (1) and (2). 



