5. Given $x \geq 1, y \geq 1$ and $\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}$, then the maximum value of $u=\lg x y$ is $\qquad$

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Answer $2+2 \sqrt{2}$.
Analysis From the given, $\lg ^{2} x-2 \lg x+1+\lg ^{2} y-2 \lg y+1=4$, so $(\lg x-1)^{2}+(\lg y-1)^{2}=4$.
Since $x \geq 1, y \geq 1$, then $\lg x \geq 0, \lg y \geq 0$, let $s=\lg x, t=\lg y$, so the point $(s, t)$ lies on the arc of a circle with center $(1,1)$ and radius 2 (in the first quadrant and on the coordinate axes). By linear programming knowledge, $u_{\max }=\sqrt{2}(\sqrt{2}+2)=2+2 \sqrt{2}$.