Question

If $z^2+z|z|+|z^2|=0$, then the locus of $z$ is

• A. A circle
• B. A straight line
• C. A pair of straight lines
• D. None of these

Answer 1

Answer: (C)

A pair of straight lines

$z^2+z|z|+|z{|}^2=0$

$\Rightarrow {\left(\frac{z}{|z|}\right)}^2+\frac{z}{|z|}+1=0$
$\Rightarrow \frac{z}{|z|}=\omega ,{\omega }^2$

$\Rightarrow z=\omega |z|$ or $z={\omega }^2|z|$
$\Rightarrow x+iy=|z|(\frac{-1}{2}+\frac{i\sqrt{3}}{2})$ or $x+iy=|z|(\frac{-1}{2}-\frac{i\sqrt{3}}{2})$
$\Rightarrow x=-\frac{1}{2}|z|,y=|z|\frac{\sqrt{3}}{2}$ or $x=-\frac{|z|}{2},y=-\frac{|z|\sqrt{3}}{2}$
$\Rightarrow y+\sqrt{3}x=0$ or $y-\sqrt{3}x=0$

$\Rightarrow y^2-3x^2=0$