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

The correct option is C A pair of straight lines

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