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# Given $z^2+z|z|+|z^2|=0$ the locus of z is

$\begin{array}{1 1}(A)\;\text{A straight line}&(B)\;\text{A circle}\\(C)\;\text{A pair of straight line}&(D)\;\text{None of these}\end{array}$

Let $z=r(\cos \theta+i\sin \theta)=re^{i\theta}$
$z^2=r^2e^{i2\theta}$
$z^2+z|z|+|z^2|=0$
$\Rightarrow r^2e^{2i\theta}+r^2e^{i\theta}+r^2=0$
$\Rightarrow (e^{i\theta})^2+e^{i\theta}+1=0$
$\Rightarrow e^{i\theta}=\large\frac{-1\pm \sqrt {3i}}{2}$