Find the shortest distance between $\begin{align}
&\frac{x\_1^2}{a\_1^2}+...+\frac{x\_n^2}{a\_n^2}=1\\
&\frac{x\_1^2}{a\_1^2}+...+\frac{x\_n^2}{a\_n^2}=2
\end{align}$.

My thinking is when $n=2$, we have two ellipses where the axes of the second are the same as the first scaled by a factor of $\sqrt{2}$. WLOG, assume $|a\_1|<|a\_2|$. Then the shortest distance would be $|a\_1|(\sqrt{2}-1)$. For the ellipsoid case, I would say the shortest distance is $|a\_{\text{min}}|(\sqrt{2}-1)$. I don't know how to prove this formally. I thought about using Lagrange multipliers but had difficulties even setting it up. Any suggestions?