10. When $x, y, z$ are positive numbers, the maximum value of $\frac{4 x z+y z}{x^{2}+y^{2}+z^{2}}$ is $\qquad$ -
$$
x^{2}+y^{2}+z^{2}=x^{2}+\frac{16}{17} z^{2}+y^{2}+\frac{1}{17} z^{2} \geqslant \frac{8 x z}{\sqrt{17}}+\frac{2 y z}{\sqrt{17}}=\frac{2}{\sqrt{17}}(4 x z+y z) \text {, }
$$

Therefore, $\frac{4 x z+y z}{x^{2}+y^{2}+z^{2}} \leqslant \frac{\sqrt{17}}{2}$, equality holds when $x=\frac{4 z}{\sqrt{17}}, y=\frac{z}{\sqrt{17}}$.