ANALYTIC QEOMKli;\ 



69. Equations of straight lines: polar coordinates. 



(1) Line through two <jivcn points. Lrt OR be the initial 



line, the pole, PI 

 = (pi, 0i ), and P 2 = 

 (p*^), tin t\vo given 

 points, and let P = 

 (p, 6) be any other 

 ^- point on the lino 

 through P, and /" 



Then (if A stands for 'area of triangle') 

 A 0P,P 2 = A 0P,P + A OPT 



sn - 



hence 



sin (^ 6\ 



sn ~ 



[21] 



This equation may also be written in the form 

 sin (O l - 2 ) sin (0 2 0) sin (6-6 



0.* 



(2) Equation of the line in terms of the perpendicular upon 

 it from the pole, and the angle which this perpendicular makes 

 with the initial line. Let OR be the initial line, the pole, 

 and LK the line whose equation is 

 sought. Also, let N= (p, a) be the 

 foot of the perpendicular from 

 upon LK, and let P=(p, 6) be any 

 otlu-r point on LK. Draw ON and 

 OP ; then 



||=cos NOP, 



i.e., p cos (0 a) = jo, 



which is the required equation. 



Observe the symmetry here ; cf. foot-note, Art. 29. 



