Question

Locate the complex number $z=x+iy$ for which $\log \sqrt{3}\left(\frac{|z{|}^2-|z|+1}{2+|z|}\right)<2$. The locus is inside a circle of radius $k$ with centre at the origin. Find the value of $k$.

Answer 1

$\log \left(\frac{|z{|}^2-|z|+1}{2+|z|}\right)<3$
$\Rightarrow \frac{|z{|}^2-|z|+1}{2+|z|}<3$
$\Rightarrow |z{|}^2-4|z|-5<0$
$\Rightarrow (\left|z\right|+1)(\left|z\right|-5)<0$
Since, $\left|z\right|+1>0$
Therefore, $\left|z\right|-5<0$
$\Rightarrow k=5$