Solution
= Solution
Using the <complex conjugate>, $b\bar z+\bar bz=2\operatorname{Re}(\bar bz)$. The <locus> is therefore the <straight line>
$$
\boxed{\operatorname{Re}(\bar bz)=-\frac c2}.
$$
In the <complex plane>, the quantity $\operatorname{Re}(\bar bz)/|b|$ is the signed <scalar projection> of $z$ in the direction $b$. Thus the <straight line> is perpendicular to $b$ and passes through $-cb/(2|b|^2)$. This description uses no separation of $z$ into real coordinates.