40.) 



LOCUS or AH JPQ' 



reaenU 



tmitflit In and e<j 



tht hue A H. bisecting re- 



axtw. 1 ue shown that the 



eqmn 



- y ).0. (8) 



\\ l..lt is til. - y* = " 



foiinr.1 troin equations (1) and < 

 has for its locus U-ti. ii.< - lim-s. 



/'roof. If /', ) in any point on CD, then 



nates satisfy equation (1), hence z l + y, 0, and there- 

 j yj)= <> : wliich shows that P l is a point 

 < locus of <<<]uatioii (:',). Hut since P, was any point 

 />. tli rrfoi.- the coordinates of every point on CD sat 



<., all jKjinte of CD belong to the locus of 

 rquation (8). 



In t)u same way it is shown that AB belongs to the 



is of equati< 



Moreover, if <P s = (jr s , y s ) be any point not on AB nor 

 on CD, then x^ + y f ** 0, and x, y t ^ 0, hence 



/* t does not belong to the locus of equation (3). 



nee the locus of equation (8) contains the loci of equa- 



K but contains no otlu i joints. 



The above theorem may be stated briefly thus : if u, r, w, 

 be any fmu-ti >ns of two variables, tlien the equation 

 vip**-*aO has for its locus al.m.-.i ;.,,i a f the 



equations u = 0, 9 - 0, w 0, etc. 



NOTE. When pootible, factoring the first member of an equation, 

 whom necond member is sero, simplifies the work of finding the locus of 

 the given equal i 



