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?

Hint: given any two points $z_1, z_2 \in \mathbb{C}\,$, then $z$ is collinear with $z_1, z_2$ iff there exists $\lambda \in \mathbb{R}$ such that $z-z_1 = \lambda(z-z_2)$. Eliminate $\lambda$ between the following, then define $z_0, D$ appropriately:
$$
\begin{cases}
\begin{align}
z-z_1 &= \lambda(z-z_2) \\
\bar z- \bar z_1 &= \lambda(\bar z- \bar z_2)
\end{align}
\end{cases}
$$
