How to find the locus of the complex equation :
$z \overline{z} +az +b\overline{z}+c=0$
I have no clue how to find the locus of such equation in complex plane. Please guide on this thanks in advance....
2026-04-17 20:02:50.1776456170
How to find the locus of the complex equation : $z \overline{z} +az +b\overline{z}+c=0$
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Let $z=x+iy$, then $z\overline z = x^2+y^2$ so, assuming $a,b,c\in\mathbb R$, our equation splits into the real part $$ x^2+y^2 + (a+b)x + c = 0 $$ and the imaginary part $$ (a- b)y = 0. $$ If $a\neq b$ this implies $y=0$ and you are left with a quadratic equation in $x$.
If $a=b$ the second equation is always true and the first equation becomes $$ x^2+y^2 + 2ax + c =0. $$ Completing the square for $x$ gives you $$ (x+a)^2+y^2 = a^2 - c $$ which is the equation of a circle centered at $-a$.