110 .IN I A)'//' /'//I- [Cli. \. 



Geuiiu-trirally it is well known that two such hi <,! ,,rs, 



(8) and < 1 i. .no perpendicular to eacli other: their -jua- 



tinns ul th.it 1'arl. 



The equations <>f the bisectors of the singles between any 

 two lines, M . 1 jr 4- B^y + C l = ami , I tf r + B^y + C a = 0, 

 are found in preriselx tin- saint- ua\ as that cniployol in tin- 

 numerical example just considered. 



EXERCISES 



1. Find tin* Aquations (f tin- bisectors of the angles between the two 

 lilies x - y + 6 = and ? * ~ 4 = f>y -7. 



2. Show that the line lljr + 3y + l=0 bisects one of the angles 

 between the two lines 12;r-5y-f7 = 0, and 3x + 4y-2 = 0. Which 

 angle is it? Kind the equation of the bisector of the other angle. 



3. Show analytically that the bisectors of the interior angles of the 

 triangle whose vertices are the points (1, 2), (.1, 3), and (4, 7) m> < t in ;i 

 otnmon point. 



4. Show analytically, for the triangle of Kx. 3, that the bisectors of 

 one interior and the two opposite exterior angles meet in a common 

 point. 



5. Find the angle from the line 3ar + y + 12 = 0to the linn nr + by 

 + 1 = 0, and also the angle from the line ax + by + 1 = to the line 

 z + 2y - 1 =0. 



By imposing upon a and b the two conditions: (1) that the angles 

 just found are equal, and (2) that the line ax + by + 1 = passes tin 

 the intersection of the other two lines, determine a and b so that this line 

 shall be a bisector of one of the angles made by the other two . 

 lines. 



66. The equation of two lines. By the reasoning given in 

 Ait. 40, it is shown that if two straight lines are represented 



by the equations 



A lX + B$ + C l = . . . (1) 



and A& + B# + , = 0, . . . (2) 



then both these lines are represent. <1 )>y the equation 



-- CiX^a: + J^ + C 2 ) = 0; . . .(3) 



