jmc

Clearly $C$ must be the midpoint of $AB$ if $C$ is the closest point to both $A$ and $B.$ Using the midpoint formula, we see that: $$\left(\frac{1+x}{2},\frac{8+y}{2}\right)=\left(3,5\right).$$Therefore, $\frac{1+x}{2}=3$, so $x=5.$ Meanwhile, $\frac{8+y}{2}=5$, so $y=2$. Our answer is $xy=\boxed{10}.$