GEOMETRY 47 



2.412 Parametric Equations of Ellipse, 



x = a cos </>, y = b sin <. 



</> = angle XOP', where P' is the point where the ordinate at P meets the 

 eccentric circle, drawn with O as center and radius a. 



2.413 OF = OF' = ea 



Va 2 - b 2 

 e = eccentricity = > 



FL = = a(i e 2 ) = semi latus rectum. 

 a 



F'P = a + c, FP = a- ex, FP + F'P = 20. 

 T = angle *rr. 

 bx 



tan T = 



aVa 2 - x 2 



NM = v - , ON = e 2 *, or = -, or = -, 



OS = . 



Va 2 - e 2 x 2 



2.414 DD' parallel to T'T; DD' and PP f are conjugate diameters: 



OD 2 = a 2 - e 2 x 2 = FP X F'P. 

 OP 2 + OD 2 = a 2 + 6 2 . 



PSxOD = ab. 

 Equation of Ellipse referred to conjugate diameters as axes: 



x 2 y 2 OL = angle XOP 



a/ 2 ^ b^ = T ft = angle XOD 



a' = OD' 

 b f =OP 



a 2 b 2 



a 2 sin 2 a + b 2 cos 2 a 

 a 2 b 2 



tan a tan & = - - 



a 2 sin 2 /3 + b 2 cos 2 

 2.415 Radius of curvature of Ellipse: 



angle FPN = angle F'P^V = w, 



eay 

 tan co = -rf , 



_ = _ + _- 



p cos co FP^ F'P' 



