M 



ANALYTIC GEOMETRY 



<... IV. 



Substituting for M V P V MP, OM V OM< and an-l.- A'/ 1 ,/ 1 

 their values, and remembering that tan 30 =t = 



this njiiiitinn Iwcomes 



0. 



The equation just found is satisfied by the coordinates of 

 any point on the given line, but is not satisfied by the coor- 

 dinates of any point that is not on this line (cf. Art. 43); 

 hence it is the equation of the line (cf. Art. 35). 



45. Equation of a circle; polar coordinates.! In deriving 

 this equation, let polar r<,,,rdi nates be employed, merely for 



variety, and let the pole be tak< -n 

 on the circumference, with a di- 

 ameter OA extended for the ini- 

 tial line. Let P = (p, 0) be any 

 point on the circle,^ and let r be 

 the radius of the circle. 



Connect P and A by a straight 



i, ..:.!. 



The positive side of this line is that side on which the origin lies (cf. 

 foot-note, Art. 43). 



t See also Art. 88. 



' xcept in elementary geometry, the word " circle " i employed by most 

 \vriu-rs on mathematics to mean "circumference of a circle." It will be so 

 used in this book. 



