L 1 N 



2. Required the equation to a straight line passing through a given 

 point, whose co-ordinates are a", y'. 



Any point of which the co-ordinates are x, y being assumed in the 

 line, we have y a x + b ; also y 1 - a x' -f b ; /. equation required is 

 y y - a (# #') 



For the sake of brevity it is usual to designate the point, whose co- 

 ordinates are #', y' y as the point (#', y') ; and the straight line, whose 

 equation is y ax -4- b, as the straight line y = a x -f- b. 



2. Required the equation to the line which passes through two given 

 points (,f, y'} and (-1"', #"). 



3. Required the angle formed by the intersection of two given lines. 

 Let y a x -\- b and y = a'x 4. b' be the given lines, and the given 

 angle ; then 



a' 



tan. 6 = 



1 +aa' 

 a a 1 



1 +' 



the positive sign being used when the / is acute, the negative when it 

 is obtuse. 

 4. Required the equation to a straight line drawn through a given 



- i 

 point (#', y'}, and making an angle tan. m with the line y = a x + b. 



TT am, 



Here y y' = - (* ^')- 



l _|_ am 



Hence (1 ) when the lines are perpendicular, 



(2) When they are parallel, 



yy> = a(x A"). 

 5. Required the distance (r) between two points (x> y] and (.r', /'). 



" v} + (./' - ?/}* 1 



When a," arid //' o, r ^ (.r* -|- yi] ; which therefore expre*se the 

 distance of a point from the origin. 



167 K 2 



