10. If the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ always passes through the fixed point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, and the length of its major axis is in the range $[\sqrt{5}, \sqrt{6}]$, then the range of the eccentricity of the ellipse is $\qquad$
$$
\begin{array}{l}
\frac{1}{a^{2}}+\frac{1}{b^{2}}=2 \Rightarrow \frac{b^{2}}{a^{2}}+1=2 b^{2} \Rightarrow 2-e^{2}=2 a^{2}\left(1-e^{2}\right) \Rightarrow \frac{2-e^{2}}{1-e^{2}}=2 a^{2} \in\left[\frac{5}{2}, 3\right] \\
\Rightarrow \frac{1}{1-e^{2}} \in\left[\frac{3}{2}, 2\right] \Rightarrow e^{2} \in\left[\frac{1}{3}, \frac{1}{2}\right] \Rightarrow \frac{\sqrt{3}}{3} \leqslant e \leqslant \frac{\sqrt{2}}{2} .
\end{array}
$$