179 



ii no: i : I.IP8B 

 Special Equstion of the Second Degree 





109. The ellipse defined. An ellipse U the locus of a 



\\ln.-ii moves so th.it th- ratio ,,f iu distance f 



a fixed jmint, culled the focus, t< its li stance fr-.m .1 li\ l 

 line, called the directrix, in constant and l->s titan unity. 

 The constant ratio is called thu eccentricity <>f the ellipse. 

 oonio section with eccentricity t<\. 

 -.) 

 .ju.it i..ii of an ellipse \vitli uny given focus, - 



- * i riitririt \ may be readily obtained from this definition. 



VMI-I.E. An ellipse of eccentricity | luw iU foctu n- at;d 



hM the line x + 2y - 5 for dire* /= (r, y) (1 



the curve, F the focus, and PQ the perpendicular from /' to 

 the direct: 



P = V(ap-2)-Kf4.1), QP = (ArU.26,W), 



+ V6 



(y + 1) = A (* -I- 2y - 5); 

 that is, 41 x - 16xy + 29 y* - 140* -I- 170y -f- 125 = 0; 

 It is the equation of the given ellipse. 



AM in the case of the parabola, so also here, a particular 



ice <>f the roonlinate axes gives a simpler form for 

 equation of the ellipse ; an equation which is more suitable 

 h. xtu.lv of the curve, and to which every equation 

 esenting an ellipse can be reduced. As has been seen in 

 K the curve is symmetrical with respect to the line 

 through the focus and perpendfalllai to the direetrix: and 

 1 lim- in t\\. otu- ..n .-ither side of tl 



ellipse will be iu a si inkier furm if this 



