What is the complex number equation $a\overline z+\overline az+b=0$ used for?

Question

I keep seeing this equation as the general equation of a line for complex numbers, but what sort of a line is it? What is the point of learning this particular equation? This is not a duplicate of another question about $a\overline z+bz+c=0$ since $\overline a$ is the conjugate of $a$ and $\overline z$ is a conjugate of $z$. Would someone please tell me just what this equation is used for and what type of problems it solves?

Answer 1

since two complex numbers are equal iff both of their real and imaginary parts are equal then if you set $z= z_x+iz_y,\; a=a_x+ia_y,\;b=b_x+ib_y$

you get a system of equation that is equivalent to your initial equation :

$$\begin{align} a_xz_x+a_yz_y & = -\frac{b_x}{2} \; \text{[notice how it looks like a dot product]} \\ b_y &= 0 \\ \end{align}$$

also recall that if you see $z$ and $a$ as points in the plane then :

$a_xz_x+a_yz_y = |a|\cdot|z| \cos \theta$

so you can say given some $b$ a possible interpretation of the equation is that it discusses the angle between $z$ and $a$ which is $$\theta = \arccos[ -\frac{b_x}{2}(|a|\cdot|z|)^{-1}]$$