10. If for all positive real numbers $x, y$, we always have
$$
\frac{x y}{\sqrt{\left(x^{2}+y^{2}\right)\left(3 x^{2}+y^{2}\right)}} \leqslant \frac{1}{k} \text {. }
$$

then the maximum value of $k$ is $\qquad$
$10.1+\sqrt{3}$