268 ANALYIir <,1<>METRY [Cii \1. 



{a) Construction by separate points. Let A' A be the given 

 difference of the focal distances, i.e., the transverse axis 

 of the hyperbola, and F^ and F 3 the given fixed points, 



the foci. With cither 



/' 



focus, say F^ as a center, 

 and a radius A' R > A '.!. 

 r * * /' 1- scribe an arc ; tlicn 

 Pio.ns. -.1--' \\i\\i the other focus as 



a center, and a radius 



AR describe an arc cutting the first arcs in tin- two points 

 /V These are points of the hyperbola. Similarly, as many 

 points as desired may be obtained and then connected by a 

 smooth curve, approximately an hyperbola. 



() Construction by a continuously moving point ; the 



i given. Pivot a straight edge LM at one focus F v so 

 that F^M is greater than the trans- 

 verse axis 2 a ; at M and the other 

 focus FI fasten the ends of a string 

 of length I, such that F 1 M=l+ 2 a ; 

 then a pencil P held against the ^ 

 string and straight edge (see Fig. 

 114), so as to keep the string always taut, will, while tin; 

 straight edge revolves about F v trace one branch of tl it- 

 hyperbola. By fastening the string at the first focus and 

 the straight edge at the second, the other branch of the curve 

 can be traced. 



163. The tangent and normal bisect internally and exter- 

 nally the angles between the focal radii of the point of contact. 



Let F\ and F t be the foci of the hyperbola V- = \i 

 P V T the tangent, and PiN the normal at the point 



