Kquation (X ) slm\\s tliat tan \\ill have 



imaginary valnee, if J5T 2 AB<Q; 



two real and coincident values, if ff 2 AB = ; 

 two real and distinct values, if J5T 2 AB>0. 



Therefore, then' is no direction, one direction. >r there 

 are two direction-. respectively, in which a line meeting 

 tlic curve in an iniinitely distant point may be dra\\n 

 through tlic origin, according ;|S 



H* - AB is < 0, = 0, or > ; 

 and hence, 



it fr 2 AB<Q, equation (1) represents an ellipse, 

 if IP AB 0, equation (1) represents a parabola, 

 if H* AB>Qi equation (1) represents an hyperbola. 



178. Center of a conic section. As already defined ( Arts. 

 111. 117, 120), the center of a curve is a point such that all 

 chords of the curve passing through it are bisected by it. 

 It has also been shown that such a point exists for the 

 ellipse and hyperbol hat these are central conies. 



If the equation of the conic is given in the fnnn 



A** + 2ffxy + By* + 2Gx+2Fy+C=0, . (1) 



the necessary and snflicient condition that the origin is at 

 the center, is Q- = and F = 0. 



1 >r if the origin be at the center, and (x r y^) be any 

 given point on the locus of equation (1), then (~x v //, ) 

 must also be on this locus (because these two points are on 

 a straight line through the origin and equidistant from 

 hence the coordinates of each of these points satisfy equa- 

 tion ( 1 >. 

 t.e. f Ax* + 2 Hx^ + By* + 2 ^ -I- 2 Fy l + C = 0, . (2) 



