Question

If $z$ is a complex number that satisfies $z^2+z|z|+|z{|}^2=0,$ then the locus of $z$ is

• A. a circle
• B. a straight line
• C. a pair of straight lines
• D. an ellipse

Answer 1

The correct option is C a pair of straight lines

$z^2+z|z|+|z{|}^2=0\Rightarrow {\left(\frac{z}{|z|}\right)}^2+\frac{z}{|z|}+1=0$
Assuming $\frac{z}{|z|}=t$
$\Rightarrow t^2+t+1=0\Rightarrow t=\omega ,{\omega }^2\Rightarrow \frac{z}{|z|}=\omega ,{\omega }^2\Rightarrow z=\omega |z|\text{or}z={\omega }^2|z|\Rightarrow x+iy=|z|(-\frac{1}{2}\pm \frac{i\sqrt{3}}{2})\Rightarrow x=-\frac{|z|}{2},y=\pm \frac{\sqrt{3}|z|}{2}$
Squaring both the sides,
$4x^2=x^2+y^2,4y^2=3(x^2+y^2)\Rightarrow 3x^2-y^2=0$

This represents a pair of straight lines.