94 AIf.li.vnr 9ROMXTBT [Cii. V 



as before. These, then, are the values of p, sin a. and 

 which are to be substituted in x cos a + y sin a = jt?. 



n 





Hence 



is an equation representing the same locus as Ax + By + (7=0, 

 and having the normal form. 



59. To trace the locus of an equation of the first degree. In 

 Art. 57 it was proved that the locus of an equation of the 



first degree in two variables is a 

 straight line; but a straight line- 

 is fully determined by any two 

 points on it; hence, to trace the 

 locus of a first degree equation it 

 is only necessary to determine t\\. 

 of its points, and then to draw tin- 

 indefinite straight line through them. The two points most 

 easily determined, and plotted, are those in which the locus 

 cuts the axes ; they are therefore the most advantageous 

 points to employ. If the line is parallel to an axis, then 

 only one point is needed. 



E.g., to trace the locus of the equation 



the ordinate of the point in which this line crosses the z-axis 

 is ; let its abscissa be x r then (x r 0) must satisfy the equa- 

 tion 2a;-3y + 12 = 0; 



hence 2x l 3 + 12 = 0, 



whence x l = 6, 



t.., the line crosses the ar-axis at the point (~6, 0). In like 

 manner it is shown that it crosses the y-axis at the point 

 (0, 4). Therefore LM is the locus of 2 x - 3 y + 12 = 0. 



