10. Let the eccentricities of an ellipse and a hyperbola with common foci be $e_{1}$ and $e_{2}$, respectively. If the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, then the maximum value of $\frac{1}{e_{1}}+\frac{1}{e_{2}}$ is $\qquad$.
10. $\frac{5}{2}$.

Let the imaginary semi-axis length of the hyperbola be $b$. Then the minor semi-axis length of the ellipse is $2b$.
Assume the foci are $(\pm c, 0)(0 < b < c)$. Then
$$
\frac{1}{e_{1}}=\frac{\sqrt{c^{2}+(2 b)^{2}}}{c}, \frac{1}{e_{2}}=\frac{\sqrt{c^{2}-b^{2}}}{c} \text {. }
$$

Thus, $\frac{1}{e_{1}^{2}}+\frac{4}{e_{2}^{2}}=5$.
By the Cauchy-Schwarz inequality, $\left(\frac{1}{e_{1}^{2}}+\frac{4}{e_{2}^{2}}\right)\left(1+\frac{1}{4}\right) \geqslant\left(\frac{1}{e_{1}}+\frac{1}{e_{2}}\right)^{2}$. Therefore, $\frac{1}{e_{1}}+\frac{1}{e_{2}} \leqslant \frac{5}{2}$, with equality holding if and only if $e_{1}=\frac{1}{2}, e_{2}=2$.