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Q. If $k + \left |k + z^{2}\right | = \left |z\right|^{2} \left(k \in R^{-}\right),$ then possible argument of $z$ is

Complex Numbers and Quadratic Equations

Solution:

$|k + z^{2}| =|z^{2} | - k = |z^{2}| + |k|$
$\Rightarrow k, z^{2}$ and $0 +i0$ are collinear
$\Rightarrow $ arg $(z^{2}) =$ arg $(k)$
$\Rightarrow 2$ arg $(z) =\pi$
$\Rightarrow $ arg $(z) = \frac{\pi}{2}$