Word Problems Using Ellipses

In an attempt to mow his lawn more efficiently, Will ties a 20-meter rope to two posts in the ground, spaced 12 meters apart. He then loops the rope around his lawn mower. Once turned on, the mower traces out a horizontal ellipse. If the equation of this ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then what are $a$ and $b$?