10. As shown in Figure 3, from the left focus $F$ of the hyperbola $\frac{x}{a^{2}}-\frac{y}{b^{2}}=1(a>0, b>0)$, a tangent line is drawn to the circle $x^{2}+y^{2}=a^{2}$. The point of tangency is $T$, and the extension of $F T$ intersects the right branch of the hyperbola at point $P$. If $M$ is the midpoint of line segment $F P$, and $O$ is the origin, then the relationship between $|M O| - |M T|$ and $b-a$ is ( ).
(A) $|M O| - |M T|$
$$
>b-a
$$
(B) $|M O| - |M T| = b-a$
(C) $|M O| - |M T| < b-a$
(D) Uncertain
10.B.

As shown in Figure 8, let the right focus of the hyperbola be $F^{\prime}$, and connect $P F^{\prime}$, $O T$. In the right triangle $\triangle O T F$, given $|F O|=c,|O T|=a$ (where $c$ is the semi-focal distance of the hyperbola), we get $|T F|=b$. Therefore, according to the midline theorem of triangles and the definition of a hyperbola, we have
$$
\begin{array}{l}
|M O|-|M T|=\frac{1}{2}\left|P F^{\prime}\right|-\left(\frac{1}{2}|P F|-b\right) \\
=b-\frac{1}{2}\left(|P F|-\left|P F^{\prime}\right|\right)=b-a
\end{array}
$$