10. Given point $P$ is on the line $x+2 y-1=0$, point $Q$ is on the line $x+2 y+3=0$, the midpoint of $P Q$ is $M\left(x_{0}, y_{0}\right)$, and $y_{0}>x_{0}+2$. Then the range of $\frac{y_{0}}{x_{0}}$ is
10. $\left(-\frac{1}{2},-\frac{1}{5}\right)$.

Notice that, the two lines are parallel.
Therefore, the locus of point $M$ is a line parallel to the two lines and equidistant from them, with the equation $x+2 y+1=0$, i.e., $M\left(x_{0}, y_{0}\right)$ satisfies $x_{0}+2 y_{0}+1=0$.

The points satisfying the inequality $y_{0}>x_{0}+2$ are in the upper left of the line $y=x+2$.
Thus, the problem is transformed into finding the range of $\frac{y_{0}}{x_{0}}$ for the point $M\left(x_{0}, y_{0}\right)$ on the ray
$$
x_{0}+2 y_{0}+1=0\left(x_{0}<-\frac{5}{3}\right)
$$

Hence, $\frac{y_{0}}{x_{0}}=k_{O M} \in\left(-\frac{1}{2},-\frac{1}{5}\right)$.