Equation of a line on a complex plane

complex-numbers

Equation of a line parallel to the line $z\bar{a} + \bar{z} a + b = 0$ is $z\bar{a} + \bar{z} a + c = 0$, (where c is a real number)

Equation of a line perpendicular to the line $z\bar{a} + \bar{z} a + b = 0$ is $z\bar{a} + \bar{z} a + c\imath= 0$, (where c is a real number)

How is the second one possible when that constant part should be a real number in an equation of a line.

I'm just starting with geometry of complex numbers, please use less rigorous things in your answers.


The line perpendicular to the line $z \bar{a} + \bar{z}a + b =0$ will be given by equation $(i z) \bar{a} + (\overline{iz})a + c =0$ with arbitrary $c\in\mathbb{R}$, which, using the fact that $\bar{i}=-i$, you can write as $$ i(z \bar{a} - \bar{z}a) + c =0 $$ $$ z \bar{a} - \bar{z}a -i c =0 $$

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