ANALYTi' r;/;o.v /://;)' [('... vui. 



, 1 ' 

 x 1:1 



tin 1 method illustrated in the numerical example. 



the general equation (1) may he written in the lunn 



(2) 



A 



wherein (cf. Art. 113, p. 188), 



BG*+AF*-ABC 



~ 



Since A and J5 have opposite signs, the two terms in the 

 first member <>f this equation are of opposite signs; tin- 

 equation is therefore in the form of [48] or [49], and repre- 

 sents an hyperbola. Its axes are parallel to the coordinate 



axes, its center is the point ( - , \ and its semi-axes 



- 



NOTE. Since A and B have opposite signs, equation (2), whirli is 

 only another form of equation (1), always represents a real locus; it i- .in 

 hyperbola proper except when A BC = BG* + AF 2 , and it then represent* 

 a pair of intersecting straight lines (cf . Art. 67). 



It is clear that the method shown for the ellipse in A it . 1 11 

 can be applied equally well to the hyperbola, to reduce ;m\ 

 equation of this curve to the standard form, when the direc- 

 trix is known. The problem of reducing to the standard 

 form the general equation of an hyperbola, when the direct rix 

 and focus are not known, is considered in full in Chapter XII. 



* That sign (+ or ) which makes the fraction positive is to be i 



