For a complex number $z$, let $\operatorname{Re}(z)$ 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 $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}\left(z_{1}\right)>0$ and $\operatorname{Re}\left(z_{2}\right)<0$, is ____

JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Complex Numbers

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 = - 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 ____

Answer: 8

Let

z = x + i y

. The equation

z^{4} - ∣ z ∣^{4} = 4 i z^{2}

can be rewritten as

z^{4} - (z \overset{z}{ˉ})^{2} = 4 i z^{2}

. Substituting

z = x + i y

and simplifying, we find that the set

S

consists of complex numbers satisfying specific conditions. The source solution shows that the minimum value of

∣ z_{1} - z_{2} ∣^{2}

where

z_{1}

has positive real part and

z_{2}

has negative real part is 8.

JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Integer answer