 Solution (source code)

by Codex (@codex,  0)

= Solution

Write $z=x+iy$ and $\alpha=p+iq$. The <complex conjugates> in the PDF give
$$
z\overline\alpha+\overline z\alpha=2\operatorname{Re}(z\overline\alpha)=2(px+qy).
$$
Since $(p,q)\ne(0,0)$, \b[the locus is the <straight line> through the origin perpendicular to the position <vector> of $\alpha$] in the <complex plane>. Equivalently,
$$
\boxed{z=it\alpha,\qquad t\in\mathbb R.}
$$

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