jmc

By the distance formula, we are trying to minimize $\sqrt{x^2+y^2}=\sqrt{x^2+\frac{1}{4}{x}^4-9x^2+81}$. In general, minimization problems like this require calculus, but one optimization method that sometimes works is to try to complete the square. Pulling out a factor of $\frac{1}{4}$ from under the radical, we have $$\begin{array}{rl}\frac{1}{2}\sqrt{4x^2+x^4-36x^2+324}& =\frac{1}{2}\sqrt{(x^4-32x^2+256)+68}\\ & =\frac{1}{2}\sqrt{(x^2-16{)}^2+68}\end{array}$$This last expression is minimized when the square equals $0$, that is, when $x^2=16$. Then the distance is $\frac{\sqrt{68}}{2}=\sqrt{17}$. Hence the desired answer is ${\sqrt{17}}^2=\boxed{17}$.