How do you determine the slant asymptotes of a hyperbola?

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Answer 1 · 2014-11-10T17:17:41

Let us find the slant asymptotes of a hyperbola of the form

#x^2/a^2-y^2/b^2=1#.

By subtracting #x^2/a^2#,

#=>-y^2/b^2=-x^2/a^2+1#

by multiplying by #-b^2#,

#=> y^2=b^2/a^2 x^2-b^2#

by taking the square-root,

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

For large #x#, #-b^2# in the square-root is negligible,

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

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

I hope that this was helpful.