10.3 .1 * Find the complex equation of the line passing through $W$ and perpendicular to a certain line $O A$, where the complex numbers corresponding to $W$ and $A$ are $w$ and $\alpha (\alpha \neq 0)$, and $O$ is the origin.
For points on the plane different from $O$, let the corresponding complex number be $\alpha$. Then, the equation of the perpendicular bisector of the segment joining this point and the origin is $\frac{z}{\alpha}+\frac{\bar{z}}{\bar{\alpha}}=1$, which can be proven using $|z|=|z-\alpha|$.

In general, $\frac{z}{\alpha}+\frac{\bar{z}}{\bar{\alpha}}=k$ (where $k$ is a real number, including zero), represents all lines perpendicular to the line connecting $O$ and $\alpha$. If $\alpha$ is replaced by $\alpha$, then we get $\frac{z}{\alpha}-\frac{\bar{z}}{\bar{\alpha}}=k \mathrm{i}$. Therefore, the equation of the line passing through the point with complex number $w$ and perpendicular to $\frac{z}{\alpha}+\frac{\bar{z}}{\bar{\alpha}}=k$ is
$$
\frac{z}{\alpha}-\frac{\bar{z}}{\bar{\alpha}}=\frac{w}{\alpha}-\frac{\bar{w}}{\bar{\alpha}}.
$$