Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$
Solution 1:
HINT:
$$z+\bar z=2\text{Re}(z)\implies zz_0+\bar z\bar z_0=2\text{Re}(zz_0)$$
Solution 2:
You know that a vertical straight line can be defined as $z+\bar z= D$, so if you rotate it's points with angle $\theta$ you get $(e^{i\theta}z)+ \overline{(e^{i\theta}z)}= D$ or $e^{i\theta}z + e^{-i\theta}\overline{z}= D$ and with arbitrary real $r\neq0$, $$re^{i\theta}z + re^{-i\theta}\overline{z}= rD$$ gives us $$zz_0 +\bar{z}\bar{z_0}= D_0$$ where $z_0=re^{i\theta}$ and $D_0=rD$.