Question 6 Let $P$ be a moving point on the curve $2 x^{2}-5 x y+2 y^{2}=1$. Find the minimum distance from point $P$ to the origin.
By the AM-GM inequality, we have $-2 x y \leqslant x^{2}+y^{2}$.
$$
\begin{array}{l}
\text { Therefore, } 1=2 x^{2}-5 x y+2 y^{2}=2 x^{2}+\frac{5}{2}(-2 x y)+2 y^{2} \\
\leqslant 2 x^{2}+\frac{5}{2}\left(x^{2}+y^{2}\right)+2 y^{2}=\frac{9}{2}\left(x^{2}+y^{2}\right) \\
\Rightarrow x^{2}+y^{2} \geqslant \frac{2}{9} \Rightarrow \sqrt{x^{2}+y^{2}} \geqslant \frac{\sqrt{2}}{3} .
\end{array}
$$

When $x=\frac{1}{3}, y=-\frac{1}{3}$ or $x=-\frac{1}{3}, y=\frac{1}{3}$, the equality holds.