Find equations for the asymptotes of the hyperbola x^2 divided by 9 minus y^2 divided by 81 = 1?

Answer 1

x^2/9 - y^2/81 = 1

You want to look at what happens as x and y get really big. In this case, the x^2 and y^2 terms get really, really big, so the 1 on the right hand side makes less and less of a contribution to the graph as x and y get big. So, replacing the 1 with 0 makes very little difference for large x and y.

So, the asymptotes come from the equation x^2/9 - y^2/81 = 0,
x/3 = +-y/9, y = +-3x.

This technique, seeing the relative contributions from different terms in an expression, is a good way to approach asymptote problems.

Answer 2

x^2/9 - y^2/81 = 1
Here, a = sqrt(9) = 3 and b=sqrt(81) = 9
The asymptotes are y = +/- bx/a
So y = +/- (9/3)x = +/- 3x are the two asymptotes