-J ANA I.) H' '.l.Mi:i'HY IT" \. 



The equation Aj- + By + C = V has the same locus as has 

 the equation 



(see Art. 88); but this is the ccmation (Art. 53) of a HIM- 



drawn through the point (0, =h and making \\itli tin- 



\ &/ 



x-axis the angle 6 = tan" 1 ^ J ; hence equation (3) is in 

 the required form, and 



*-~, and ^-|. 



(3) Reduction to the standard form xcosa + y sin a = p 

 (normal form). 



If equation (1) and 



x cos a + y sin a = p . . . (4) 



represent the same line, then they differ merely by some 

 constant multiplier, say k (cf. Art. 38). Then 



kAx + kBy + kC=xcosa + ysina ^ = ; 

 . . kA = cos a, kB = sin a, and kC = p ; 



whence k = 



A 



hence cos = , sma 



and p = 



If B = 0, the line represented by equation (1) is parallel to the /-axis, 

 and the slope form of the equation is inapplicable (Art. 66); but, in that case, 

 the above reduction also fails. 



