I'M 



AN A I. Y II'' 



[CH. VI1L 



the conic (3) shall, in addition t> passing through the four 

 points in which ^ = and S^ = Q intersect, satisfy OIK- other 

 rnlitiun ; <.//., that it sliall j>ass through a ,u r i\ '' lift h point . 

 Moreover, if fi^vO and X., --- <> an- hntli circles, then 

 II eircle (cf. Artfi. i.", ami 



V. POLAR EQUAT!>\ OK Till: CONIC H ' I IONS 



131. Polar equation of the conic. Hascd upon tin- "focus 

 and directrix" .lelinition already given in Art. 48, the polar 

 c<l nation of a conic section is easily derived. 



Let D*D (Fig. !'T ) ! the given line (the direct ri\) and 

 tlie nven point (the focus); draw ZOR through and per- 

 pendicular to D'D, and let l>e < -hosm 

 as the pole and OR as the initial line. 

 Also let P= fa 0) be any point on tho 

 locus, and let e be the eccentricity. 

 Draw MP and OK parallel, and LP 

 and JTK perpendicular, to />'/>, and let 

 OK=l\ then 



[definition of the curve] 

 e(ZO+ OH); 



This equation, when solved for />, may be written in the 

 form 



which is the polar equation of a conic section referred to 

 its focus and principal axis ; e being the eccentricity and I 

 the semi-lat us-rectum. If = 1, conation [55] represents a 

 parabola ; if e < 1, an ellipse j aiid if e > 1, an hyperbola. 



