HMMT 2003 G4

Farmer John is inside of an ellipse with reflective sides, given by the equation $x^{2} / a^{2}+$ $y^{2} / b^{2}=1$, with $a>b>0$. He is standing at the point (3,0), and he shines a laser pointer in the $y$-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair $(a, b)$ ?