Do you understand that it is impossible to say "where you went wrong" if you don't show what you did?
In any case, a hyperbola having the coordinate axes as axes of symmetry (any hyperbola of the form or ) has slant asymptotes, not vertical and horizontal asymptotes.
No, the anwer is not . Neither nor is correct.
Here is one way to think about it. The graph of the hyperbola gets closer and closer to the asymptotes as x and y get larger and larger. For very, very large x and y, both and are very, very large so that "1" can be ignored compared to them. That is, for very, very large x and y, , approximately. The left side is a "difference of squares" and factors as .
So we must have and , the equation of the two straight line asymptotes. Solving for y, we have as Archie Meade says.