10. For a line segment $AB$ of fixed length $m$ whose endpoints move on the right branch of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $\left(m>\frac{2 b^{2}}{a}\right)$. Then, the minimum value of the x-coordinate of the midpoint $M$ of $AB$ is (expressed in terms of $a$, $b$, and $m$).
10. $\frac{a(m+2 a)}{2 \sqrt{a^{2}+b^{2}}}$.
As shown in Figure 4, let the projections of $A$, $B$, and $M$ on the right directrix of the hyperbola be $A^{\prime}$, $B^{\prime}$, and $M^{\prime}$, respectively, and the right focus be $F$, with the eccentricity being $e$.
By the definition of the hyperbola, we have
$$
\begin{array}{l}
\left|M M^{\prime}\right|=\frac{\left|A A^{\prime}\right|+\left|B B^{\prime}\right|}{2} \\
=\frac{1}{2}\left(\frac{|A F|}{e}+\frac{|B F|}{e}\right) \\
=\frac{1}{2 e}(|A F|+|B F|) \\
\geqslant \frac{|A B|}{2 e}=\frac{m}{2 e} .
\end{array}
$$
$\therefore$ The x-coordinate of $M$ $=|M N|=\left|M M^{\prime}\right|+\left|M^{\prime} N\right|$
$$
\geqslant \frac{m}{2 e}+\frac{a^{2}}{c}=\frac{a m}{2 c}+\frac{a^{2}}{c}=\frac{a(m+2 a)}{2 \sqrt{a^{2}+b^{2}}} .
$$