Question 29
Farmer John is inside of an ellipse with reflective sides, given by the equation $x^{2} / a^{2} + y^{2} / b^{2} = 1$, with $a \gt b \gt 0$. He is standing at the point $(3, 0)$, and he shines a laser pointer in the $y$-direction. 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 $(a, b)$?