1SI 



ANALYTIC GEOMI 1 1 , 



\ III. 



I ' from m\;i pint on the ellipse a I- //////? 



to the transverse axis; then the square of the distance from the 

 center of the ellipse to the foot of this perpendicular, dii / /, / /< // 

 the square of the semi-transverse axis, plus the square of the 

 perpendicular divided by the square of the semi-conjugate axis, 

 equals unit//. 



This geometric or physic-ill property belongs to no point 

 not on the curve, and therefore completely determines tin- 

 ellipse. It enables one to write immediately the, equation of 

 any ellipse whose axes are parallel to the coordinate axes. 



For example : if, as in Fig. 82, the major axis of an (11 

 is parallel to the ./--axis, and the center is at the point 



C= (A, &), l et -P=0^y) be an 7 point on the curve, and 

 a, b be the semi-axes, then 



CM 1 NP* 

 CA* " Off" 



thatis 



which is the equation of the given ellipse. 



Or again, if, as in Fig. 83, the major axis is paralh 1 to 

 the y-axLs ; then, as before 



US' , MP 2 _, 



~r ___ -M 



CA* Off 





