180 



ANA LY IK' '.I .!/ /;//;) 



[C. 



of symmetry is rhosm as the z-axis, with the origin halt 

 way between its two points of intersection with the rum-. 

 The resulting equation is the first standard form of the < ^na- 

 tion of the ellipse. 



110. The first standard equation of the ellipse. l.-t /'!.. thr 

 liu diivrtrix, and ZFX the jnTjM-ndirnlar to It' It 



through F, cutting 

 the curve in the two 

 points A' and A 

 (Art. 48)*. Denote 

 by 2 a the length 

 of AA', and let 

 be its middle point, 

 so that 



A0= OA' = a. 

 Let ZX be the a^axis, the origin, and OY, ]< -rjx -n- 

 diriilur to OX, the y-axis. Then, by the definition ( .f the 



ellipse, 



AF= eZA, and FA' = eZA' ; 



. . AF+ FA' = e (ZA + ZA') = e(ZA + ZA + AA'), 

 i.e., AA 1 = e(2ZA + AA), 



whence 2a = 2e(ZA + 



therefore 



and tin- Aquation of the directrix is x -h - = 0. 



e 



Again, 

 i.e., 



\\licnce 



FA' - ^^= e(ZA' - ZA); 

 FO + OA' - (AO - FO) = eAA', 



Thi* equation may also be easily derived Independently of Art. 48, 

 Cf. Arts. 103, 110. 



