I'm trying to solve the following problem:
"Show that any straight line in $\mathbb R^2$ can be represented via the complex equation $\overline a z+a \overline z+b=0$ ; $a\neq 0 \in \mathbb C,b \in \mathbb R$"
"Is the choice of $a,b$ unique for each line? is it unique with the constraint $|a|=1, b\leq 0$? Give a geometric interpetation for $a$."
What I did:
It may be ugly, but I wrote $z=x+iy$ which is our variable, and $a=\alpha+i\beta$ which is our parameter.
After inputting that in $\overline a z+a \overline z+b=0$. I got $2\alpha x+2\beta y+b=0$, this is indeed a representation of a line in $\mathbb R^2$.
My problem arises in the "is this representation unique." The answer is ofcourse no, since we can multiply by whatever number we wish, for example $\alpha x+\beta y+0.5b=0$ is the exact same line as $2\alpha x+2\beta y+b=0$.
But I don't know if this representation is unique under the constraints $|a|=1, b\leq 0$. This is where I could use some assistance.
Regarding geometric interpretation, $a \in \mathbb C$ is the slope of the straight line I would assume?