Question

If $z=x+iy$, then show that $z\overline{z}+2(z+\overline{z})+b=0$, where $bϵR$, represents a circle.

Answer 1

Given : $z=x+iy$

$\therefore \overline{z}=x-iy$
Now,
$z\overline{z}+2(z+\overline{z})+b=0$
$\Rightarrow (x+iy)(x-iy)+2(x+iy+x-iy)+b=0$
$\Rightarrow [x^2-(iy{)}^2]+4x+b=0$
$\Rightarrow x^2+y^2+4x+b=0\{\because i^2=-1\}$

This represents an equation of a circle.
Hence, proved.