You're on the right track. The coefficients of (a and b) are all that matter. If a>b, it's one, and if a<b it's the other. Of course, if a=b, it's a circle.
I'll let you figure out which one is which. Look at the equation when x=0 and when y=0.
Of course, a and b have to be positive, and it's possible that you'll end up with after completing the square in x and y, so your equation would have no solutions.
Post again in this thread if you're still having trouble.