First slide

Geometry of complex numbers

Question

$For a complex number z, let Re (z) denote the real part of z . Let S be the set of all complex numbers$
$Z satisfying z^{4} - | z |^{4} = 4 i z^{2}, where i = \sqrt{- 1} . Then the minimum possible value of {|z_{1} - z_{2}|}^{2}, where$
$z_{1}, z_{2} \in S with Re (z_{1}) > 0 and Re (z_{2}) < 0, is$

Difficult

Solution

$\begin{array}{l} Let z = x + iy \\ z^{4} - | z |^{4} = 4 i z^{2} \\ \Rightarrow z^{4} - (z \bar{z})^{2} = 4 i z^{2} \\ \Rightarrow z = 0 or z^{2} - (\bar{z})^{2} = 4 i \\ \Rightarrow 4 i x y = 4 i \\ \Rightarrow x y = 1 \end{array}$
${|Z_{1} - z_{2}|}_{min}^{2} = 8$

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