ANALYTIC BMOMOnn [Cii. x. 



Again, Pff = (. *i - <*f ) 2 + //,- = <rV- -J 



-I- A, 1 , 



\ 1< nee, by addition, 



= a - ex r 



i.e., the sum of the focal distances of any point on an ellipse 

 is constant; it is equal to the major axis. 



This property gives an easy method of finding the foci ot 

 an ellipse when the axes A' A and B'B are given. 



For 



but FiO = OF* 



F 2 B=F l B = a.. 



Hence, to lind the foci, describe arcs with B as center and 

 a OA as radius, cutting A' A in the points F l ;md / 

 these points are the required foci. 



145. Construction of the ellipse. The property of Art. 

 144 is sometimes '_ r ivrn as tin- definition of the ellipse; vi/.. 

 the ellipse is the locus of a point the sum of whose *//>/./ 

 from two fixed points is constant. This definition leads at 

 once to the equation of the curve (cf. Ex. 5, p. 67); and 

 also gives a ready method for its construction. 



(a) Construction by separate point*. Let A' A be the 

 Driven sum of the fo< al distances, i.e., the major axis of tin- 

 ellipse; and F l and F 2 be the given fixed points, the foci. 

 With either focus as center, and with any radius A'R <A'A 

 describe an arc ; then with the other focus as center, and 

 radius RA* describe an arc cutting the first arc in two 

 points. These are points of the ellipse. In the same way 



