

302 RECTILINEAR ASYMPTOTES. EXAMPLES. 



Example (3). xy (y x) (?/ x + 3a) + 4a 3 .r a 4 = 0. 



Here the term containing the highest power of y is or?/ 3 ; 

 thus x = gives one asymptote, namely the axis of y. Simi- 

 larly, the term containing the highest power of x is yx 3 ; 

 therefore y = gives one asymptote, namely the axis of x. 

 Then put px + /3 for y, and we obtain the expression 



Arranging this according to powers of x, we have 

 x'fj, (/* - !) 2 + x 3 (/* - 1) ( 3 M + (3/* - 1) } 



+ x* {/3 2 (3/x-2) + 3a/3 (2/t- 1)} + ... 



Put /A (//, I) 2 = ; we have then /A = 0, or yu- = 1 ; the 

 former value of /u, will lead to the asymptote coinciding with 

 the axis of x which we have already found. The value /z = 1 

 makes the coefficient of x 3 in the above expression vanish ; 

 we therefore equate to zero the coefficient of a; 2 , putting /A = 1 

 in it. We thus obtain /3 2 -f 3a/3 = ; hence, ft = 0, or ft = - 3a. 

 Therefore we have for the equations to asymptotes y = x, and 

 y = x 3a. 



It will be observed that the conclusions of this Chapter all 

 hold whether the axes be rectangular or oblique. 



EXAMPLES. 



Find the asymptotes of the following curves : 



1. y* (x 2a) = x 3 a 3 . Result. ic = 2a; y= (x + a). 



2a 



2. / = x* (2a - x). Result, y = - x + . 



o 



3. a? + x* = a?a-x. Result. = 0. 



4. *a + lx=a?*+l?x\ Result. = -- 



5. y* = (x-af(x-c}. Result. y = x-% 



6. xy* + yx* = a 3 . Result. x = Q; 2/=0; y = -x. 



7. afy 8 = a 2 (x 2 y 2 }. Result, y = a. 



