276 AMALYll' QJEOJCMTS7 I('H. \l. 



\MIM.I: 1. An h\ jM-rl.ola lia\ing the lines 

 (1) x + 2y + 3 and (2) 3* + 4 // + r, = 

 for asymptotes, will have an equation of the form 



(a; + 2y + 3)(3:i; + 4y + 5) + Ar = 0, . . 

 while the equation of its conjugate hyperbola \\ ill be 



O + 2y + 3)(3* + 4y + 5)-* = 0. . . (4) 



If a second condition is imposed upon the hyperbola. 

 e.g.) that it shall pass through the point (1, ~1), tln-n tin 

 value of k may be easily found thus : since the curve passes 

 through the point (1, "1), therefore by equation (3), 



and the equation of the hyperbola is 



(x -f 2y + 3)(3a; + 4y 4- 5) - 8 = 0, 

 that is, 32? + IQxy + 8y 2 + 14a; + 22y + 7 = 0; . 

 and the equation of the conjugate hyperbola is 



32? + IQxy + 8/ -f 14z -f 22y -f 23 = 0. 



EXAMPLE 2. The equation of the asymptotes of tin 



hyperbola 



3ar-14a:y-5y 2 + 7^ + 13y-8 = . . . (1) 



differs from equation (1) by a constant only, hence it is of 



the form 



3s 2 -14*y-5y 2 + 7* + 13y + fc = 0. . . (2) 



Now equation (2) represents a pair of straight lines, there- 

 fore its first member can be factored, and, by Art. 67. [17] 

 _ 15 k - -LS^l _ -fc^l + l|A _ 4<>k = ; 



i.e., 64 k = - 384, whence k = 6. 



Therefore the equation of the asymptot< 



- 1 4 ry - 5 f 4- 7 x + 1 3 y - r, = 0, 



