4. 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 $(a, b)$ ?
Solution: $\square$
The points where the farmers are standing must be the foci of the ellipse, so they are $(3,0)$ and $(-3,0)$. If the total distance traveled is 10 , then $a$ must be half of that, or 5 , since the distance traveled by a ray reflecting off the wall from when it leaves one focus to when it reaches the other focus is $2 a$, the length of the major axis. If a focus is at $x=3$, then we have $3=\sqrt{a^{2}-b^{2}}=\sqrt{25-b^{2}}$, yielding $b=4$.