How do I determine the asymptotes of a hyperbola?

1 Answer
Sep 28, 2014

If a hyperbola has an equation of the form #{x^2}/{a^2}-{y^2}/{b^2}=1# #(a>0, b>0)#, then its slant asymptotes are #y=pm b/ax#.

Let us look at some details.

By observing,

#{x^2}/{a^2}-{y^2}/{b^2}=1#

by subtracting #{x^2}/{a^2}#,

#Rightarrow -{y^2}/{b^2}=-{x^2}/{a^2}+1#

by multiplying by #-b^2#,

#Rightarrow y^2={b^2}/{a^2}x^2-b^2#

by taking the square-root,

#Rightarrow y=pm sqrt{ {b^2}/{a^2}x^2-b^2 } approx pm sqrt{{b^2}/{a^2}x^2}=pm b/a x#

(Note that when #x# is large, #-b^2# is negligible.)

Hence, its slant asymptotes are #y=pm b/a x#.