10. Prove that $W=(u-v)^{2}+\left(\sqrt{2-u^{2}}-\frac{v}{9}\right)^{2}$ has a minimum value of 8 on $00$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
10. Let $P\left(u, \sqrt{2-u^{2}}\right), Q=\left(v, \frac{9}{v}\right)$, then $P$ is on the circle $x^{2}+y^{2}=2$, $Q$ is on the hyperbola $y=\frac{1}{x}$, and $W=(u-$ $v)^{2}+\left(\sqrt{2-u^{2}}-\frac{9}{v}\right)^{2}=|P Q|^{2}$, and $|P Q| \geqslant|O Q|-|O P|=\sqrt{v^{2}+\left(\frac{9}{v}\right)^{2}}-\sqrt{2} \geqslant \sqrt{2 \cdot v \cdot \frac{9}{v}}-\sqrt{2}=$ $2 \sqrt{2}$. Therefore, $W=|P Q|^{2} \geqslant(2 \sqrt{2})^{2}=8$, equality holds if and only if $O, P, Q$ are collinear, $P$ is between $O$ and $Q$, and the coordinates of $Q$ are $(3,3)$, at this time the coordinates of $P$ are $(1,1)$, so the minimum value of $W$ is 8.