10. For a line segment $AB$ of fixed length $m$ whose endpoints move on the right branch of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $\left(m>\frac{2 b^{2}}{a}\right)$, the minimum value of the x-coordinate of the midpoint $M$ of $AB$ is $\qquad$ (expressed in terms of $a$, $b$, and $m$).
10. $\frac{a(m+2 a)}{2 \sqrt{a^{2}+b^{2}}}$

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, then $| M M^{\prime} |=$
$$
\begin{array}{l}
\frac{|A A^{\prime}|+|B B^{\prime}|}{2}=\frac{1}{2}\left(\frac{|A F|}{e}+\frac{|B F|}{e}\right)= \\
\frac{1}{2 e}(|A F|+|B F|) \geqslant \frac{|A B|}{2 e}=\frac{m}{2 e}
\end{array}
$$

Therefore, the x-coordinate of $M$ $=|M N|=|M M^{\prime}|$
$$
\begin{array}{l}
+|M^{\prime} N| \geqslant \frac{m}{2 e}+\frac{a^{2}}{c}=\frac{a m}{2 c}+\frac{a^{2}}{c}= \\
\frac{a m + 2 a^{2}}{2 \sqrt{a^{2}+b^{2}}}=\frac{a(m+2 a)}{2 \sqrt{a^{2}+b^{2}}}.
\end{array}
$$