J71 ANALTII' QMOMXTRY [C* \i. 



and these equations represent the asymptotes of th> 

 l'.,l;i; they an- the lines 0& and OA'in Fig. 117. Therefore, 

 f/i // 'Hti/iH/ttotes ; they pass t1trn<ili its center, 



and are the diagonals of the rectangle described upon its axes. 

 Since the equation of the hyperbola conjugate to (1) is 



5-g=-l, ... (6) 



and thus differs from equation (1) only in the sign of the 

 second member, which affects only the constant term in 

 equation (3), therefore the equations (4) determine the 

 value of m and c fur the asymptotes of the conjugate hyp. i - 

 bola also. It follows that conjugate hyperbolas have the same 

 asymptotes. 



A second derivation of the equation of the asymptotes of an h 

 bola ( 1 ) is as follows : 



The equation of the tangent to (1) at the point (x v y^ is 



which may be written in the form 



Wr= 



Since (z p y^ is on the curve (1), 



Substituting this value of ^' in equation (8), it becomes 



which is only another form of the equation of the tangent represented 

 by equations (7) or (8). If now the point of contact (x v y,) moves 

 further and further away, so that x l = oo, then the limiting position 



the line (10) is represented by \Px a*y f - J = aby. 



Hence the equations of the asymptotes are : y = -x (cf. Art. 



a 



