9. (14 points) In the rectangular coordinate plane $x O y$, it is known that points $A$ and $B$ lie on the hyperbola $C: 2 x^{2}+4 x-y^{2}=0$, and form a right-angled isosceles triangle $\triangle A B C$ with $O$ as the right-angle vertex. Find the number of all such $\triangle O A B$.

In the rectangular coordinate plane $x O y$, it is known that points $A$ and $B$ lie on the hyperbola $C: 2 x^{2}+4 x-y^{2}=0$, and form a right-angled isosceles triangle $\triangle A B C$ with $O$ as the right-angle vertex. Find the number of all such $\triangle O A B$.
9. Let $l_{O A}: y=k x(k \neq 0)$.
Then $l_{O B}: y=-\frac{1}{k} x$.
From $\left\{\begin{array}{l}y=k x, \\ 2 x^{2}+4 x-y^{2}=0\end{array}\right.$
$\Rightarrow\left(2-k^{2}\right) x^{2}+4 x=0$
$\Rightarrow 2-k^{2} \neq 0, x_{A}=\frac{4}{k^{2}-2}$
$\Rightarrow|O A|=\sqrt{1+k^{2}}\left|\frac{4}{k^{2}-2}\right|$.
Similarly,
$$
\begin{array}{l}
|O B|=\sqrt{1+\left(\frac{1}{k}\right)^{2}}\left|\frac{4}{\left(\frac{1}{k}\right)^{2}-2}\right| \\
=\sqrt{1+k^{2}}\left|\frac{4 k}{2 k^{2}-1}\right| .
\end{array}
$$

Since $\triangle O A B$ is an isosceles right triangle with $O$ as the right-angle vertex, we have $|O A|=|O B|$.
$$
\begin{array}{l}
\text { Hence } \sqrt{1+k^{2}}\left|\frac{4}{k^{2}-2}\right|=\sqrt{1+k^{2}}\left|\frac{4 k}{2 k^{2}-1}\right| \\
\Rightarrow \frac{1}{k^{2}-2}= \pm \frac{k}{2 k^{2}-1} . \\
\text { If } \frac{1}{k^{2}-2}=\frac{k}{2 k^{2}-1} \\
\Rightarrow k^{3}-2 k^{2}-2 k+1=0 \\
\Rightarrow(k+1)\left(k^{2}-3 k+1\right)=0 \\
\Rightarrow k_{1}=-1, k_{2}=\frac{3-\sqrt{5}}{2}, k_{3}=\frac{3+\sqrt{5}}{2} . \\
\text { If } \frac{1}{k^{2}-2}=-\frac{k}{2 k^{2}-1} \\
\Rightarrow k^{3}+2 k^{2}-2 k-1=0 \\
\Rightarrow(k-1)\left(k^{2}+3 k+1\right)=0 \\
\Rightarrow k_{4}=1, k_{5}=\frac{-3-\sqrt{5}}{2}, k_{6}=\frac{-3+\sqrt{5}}{2} .
\end{array}
$$

Since $k_{1} k_{4}=k_{2} k_{5}=k_{3} k_{6}=-1$, the two triangles obtained from $k_{1}$ and $k_{4}$ are the same. Similarly, the two triangles obtained from $k_{2}$ and $k_{5}$ are the same, and the two triangles obtained from $k_{3}$ and $k_{6}$ are also the same.
In summary, there are three triangles $\triangle A B C$ that satisfy the conditions.