Question

The equation $z\overline{z}+a\overline{z}+\overline{a}z+b=0,b\in R$ represents a circle (not point circle) if

• A. ${\left|a\right|}^2>b$
• B. ${\left|a\right|}^2<b$
• C. $\left|a\right|>b$
• D. $\left|a\right|<b$

Answer 1

The correct option is A ${\left|a\right|}^2>b$

We have

$z\overline{z}+a\overline{z}+\overline{a}z+b=0$

$\Rightarrow z\overline{z}+a\overline{z}+\overline{a}z+a\overline{a}=a\overline{a}-b$

$\Rightarrow (z+a)(\overline{z}+\overline{a})=a\overline{a}-b\Rightarrow {|z+a|}^2={\left|a\right|}^2-b$

This represents a circle (not point circle) if ${\left|a\right|}^2>b$