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 z4–|z|4=4iz2, where i=−1. Then the minimum possible value of |z1–z2|2, where z1, z2 ∈S with Re( z1 ) > 0 and Re( z2 )<0, is ______

z4−|z|4=4i z2z4−z2. z¯2=4i z2⇒z2=0 or z2−z¯2=4iLet z=x+iy⇒(z+z¯)(z−z¯)=4i⇒2x . 2iy=4i⇒xy=1z1 lies on xy = 1 in first quadrant and z2 lies on xy = 1 in third quadrant.⇒ |z1 – z2|2 is minimum when z1≡ (1, 1) and z2=(–1, –1)⇒|z1 – z2|2= 8

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For a complex number $z,$ let $R e (z)$ 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 $R e (z_{1}) > 0$ and $R e (z_{2}) < 0,$ is ______

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answer is 8.

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Detailed Solution

$\begin{array}{l} z^{4} - {| z |}^{4} = 4 i z^{2} \\ z^{4} - z^{2} . {\bar{z}}^{2} = 4 i z^{2} \\ \Rightarrow z^{2} = 0 or z^{2} - {\bar{z}}^{2} = 4 i \\ Let z = x + i y \\ \Rightarrow (z + \bar{z}) (z - \bar{z}) = 4 i \\ \Rightarrow 2 x . 2 i y = 4 i \\ \Rightarrow x y = 1 \end{array}$

$z_{1}$ lies on $x y = 1$ in first quadrant and $z_{2}$ lies on $x y = 1$ in third quadrant.
$\Rightarrow | z_{1} - z_{2} |^{2}$ is minimum when $z_{1}$ $\equiv$ (1, 1) and $z_{2} = (- 1, - 1)$
$\Rightarrow | z_{1} - z_{2} |^{2} = 8$