9. (16 points) Given points $A(1,0), B(0,-2)$, and $O$ as the origin, a moving point $C$ satisfies:
$$
\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}(\alpha, \beta \in \mathbf{R}, \alpha-2 \beta=1) \text {. }
$$

Let the trajectory of point $C$ intersect the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b > 0)$ at points $M$ and $N$, and the circle with diameter $M N$ passes through the origin. If the eccentricity of the hyperbola is no greater than $\sqrt{3}$, find the range of values for $a$.
Let point $C(x, y)$. Then $(x, y)=(\alpha, -2 \beta)$.
Since $\alpha - 2 \beta = 1$, we have $x + y = 1$, which means the trajectory equation of point $C$ is $x + y = 1$.
By combining the line equation with the hyperbola equation and eliminating $y$, we get
$$
\left(b^{2} - a^{2}\right) x^{2} + 2 a^{2} x - a^{2} \left(1 + b^{2}\right) = 0.
$$

Let $M\left(x_{1}, y_{1}\right), N\left(x_{2}, y_{2}\right)$. Then
$$
x_{1} + x_{2} = \frac{2 a^{2}}{a^{2} - b^{2}}, \quad x_{1} x_{2} = \frac{a^{2} \left(1 + b^{2}\right)}{a^{2} - b^{2}}.
$$

By the problem statement,
$$
\begin{array}{l}
O M \perp O N \Rightarrow \overrightarrow{O M} \cdot \overrightarrow{O N} = 0 \\
\Rightarrow x_{1} x_{2} + y_{1} y_{2} = 0 \\
\Rightarrow x_{1} x_{2} + \left(1 - x_{1}\right) \left(1 - x_{2}\right) = 0 \\
\Rightarrow 2 x_{1} x_{2} - \left(x_{1} + x_{2}\right) + 1 = 0 \\
\Rightarrow \frac{2 a^{2} \left(1 + b^{2}\right)}{a^{2} - b^{2}} - \frac{2 a^{2}}{a^{2} - b^{2}} + 1 = 0 \\
\Rightarrow b^{2} - a^{2} = 2 a^{2} b^{2} \\
\Rightarrow c^{2} - 2 a^{2} = 2 a^{2} \left(c^{2} - a^{2}\right) \\
\Rightarrow 2 a^{2} = \frac{e^{2} - 2}{e^{2} - 1} = 1 - \frac{1}{e^{2} - 1}.
\end{array}
$$

Given $1 < e \leqslant \sqrt{3}$, we have $a^{2} \leqslant \frac{1}{4}$, hence $a \in \left(0, \frac{1}{2}\right]$.