Example 26 (2002 Anhui Provincial Competition Question) A line segment $AB$ of fixed length $m$ has its two endpoints moving 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 $\qquad$ (expressed in terms of $a, b, m$).
Solve: Fill in $\frac{a(m+2 a)}{2 \sqrt{a^{2}+b^{2}}}$. Reason:
As shown in Figure 13-1, 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
$$
\left|M M^{\prime}\right|=\frac{\left|A A^{\prime}\right|+\left|B B^{\{\prime}\right|}{2}=\frac{1}{2}\left(\frac{|A F|}{e}+\frac{|B F|}{e}\right)=\frac{1}{2 e}(|A F|+|B F|) \geqslant \frac{|A B|}{2 e}=\frac{m}{2 e} .
$$

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