Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$
I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi$ then $\bar z = x-yi$   Thus, $z+\bar z = x+yi+x-yi=2x$ which is an arbitrary vertical straight line in w-plane.
But why $zz_o + \bar z \bar z_0 = D$ is an arbitrary straight line in complex plane?

Hack the equation.
Substitute:
$$
\begin{cases}
z = x + iy   \\
z_0 = x_0 + iy_0 
\end{cases}
$$
Do some algebraic manipulations and you'll obtain the equation of a line.
Maybe what confuses you is that neither $z_0$ nor $D$ have a clear geometric interpretation in terms of intercepts, distance to the origin, etc...
Hope this helps!
