• # question_answer 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

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