194 



i \.\i.y ric GEOM /://:> 



,- ... vin. 



(8) For every value of x from a to //. y is imaginary ; 

 while for every other value of a:, y is real ami has two 



v, ilucs, equal numerically but opposite in sign, lint for 

 ; y value of y, a; has two real values, equal numerically 

 and opposite iii sign. When ./ increases numerically from a 

 to oo, then y increases also numerically from to oo. 



These facts show that no part of tin- hyperbola lies 

 Ix-tween the two lines perpendicular to its principal axis and 

 drawn through the vertices of tin- curve; but that it has 

 two open infinite branches, lying outside of these two lines. 

 The form of the hyperbola is as represented in Fig. 86. 



The segment A'A of the principal axis, intercepted by the 

 curve, is called its transverse axis. The segment B'B of i li- 



nd line of symmetry ( the 

 //-axis ), win -n- Ji'O = OB = 6, 

 is called the conjugate axis; 

 and although imt cut by 1 he 

 hyperbola, it bears impor- 

 tant relations to the curve. 

 l-'rom the symmetry of the 

 hyperbola, with respect to 

 these axes, it follows that it 

 is also symmetrical with rc- 



Fio.87 



spect to their intersection O. 



the center of the curve. It follows also that there is a 

 ond focus at the point ( ae, 0), and a second directrix in 

 the line x 4- - = on the negative side of the conjugate a 







and symmetrical to the original focus and directrix. (See 

 Art. Ill, foot-note.) 



The latus rectum of the hyperbola is readily found to be 



(cf. Arts. 105, 111). 

 a 



