10. Given that the asymptotes of a hyperbola with foci on the $x$-axis pass through the intersection points of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{16}=1$ and $\frac{a x^{2}}{16}+\frac{y^{2}}{4}=1(0<a \leqslant 1)$. Then the range of the eccentricity of the hyperbola is $\qquad$
10. $\left[\sqrt{2}, \frac{\sqrt{21}}{3}\right)$.

By solving the system of two ellipse equations, we get $y^{2}=\frac{4-a}{3} x^{2}$. Therefore, $e^{2}=1+\frac{4-a}{3}=\frac{7-a}{3} \in\left[2, \frac{7}{3}\right)$. Hence, $e \in\left[\sqrt{2}, \frac{\sqrt{21}}{3}\right)$.