(14) From the left focus $F$ of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$, draw a tangent to the circle $x^{2}+y^{2}=9$, with the point of tangency being $T$. Extend $F T$ to intersect the right branch of the hyperbola at point $P$. If $M$ is the midpoint of segment $F P$, and $O$ is the origin, find the value of $|M O|-| M T|$.
(14) Without loss of generality, place point $P$ in the first quadrant. Let $F^{\prime}$ be the right focus of the hyperbola, and connect $P F^{\prime}$. Since $M$ and $O$ are the midpoints of $F P$ and $F F^{\prime}$ respectively, we have $|M O|=\frac{1}{2}\left|P F^{\prime}\right|$. By the definition of the hyperbola, we get
$$
\begin{array}{c}
|P F|-\left|P F^{\prime}\right|=6, \\
|F T|=\sqrt{|O F|^{2}-|O T|^{2}}=4 .
\end{array}
$$

Thus,
$$
\begin{aligned}
|M O|-|M T| & =\frac{1}{2}\left|P F^{\prime}\right|-|M F|+|F T| \\
& =\frac{1}{2}\left(\left|P F^{\prime}\right|-|P F|\right)+|F T| \\
& =-3+4 \\
& =1 .
\end{aligned}
$$