116 ANALYTIC QBOMETUV ft V. 



Hence .M^-^" 1 * [law of sines] 



P l /t 8in(o>-0) 



Substituting in this equation the coordinates of P l and P, 

 it becomes 



y - ;/i = in 



ar T sin (o 0' 





which is the required equation. 



When o> = ^ this equation reduces to equation [11], 

 to y yi = m (x a?|), where w = tan ; but it must be 

 observed that if qfc3, then the coefficient of x in equation 



[18] does not represent the slope of the line. If, however, 

 the slope of the line [18], i.e., the tan 6 for this line, is 



desired, it is easily found thus : let - = &, from 



sin (to ti) 



which is obtained tan = , * f in * . 



1 + k cos to 



If, in the derivation of equation [18], the given point is 

 that in which the line LL l meets the y-axis, i.e., if PI =(0, ft), 

 then equation [18] reduces to 



sin (o>- 0) 



[19] 



which corresponds to equation [12], Imt tluj coefficient of r 

 is not the slope of the line. 



C2) Equation of a straight line in terms of the perpendic- 

 ular upon it from the origin^ and the angles which this perpen- 

 ilar makes with the axes. 



