For a complex number , let denote the real part of . Let be the set of all complex numbers satisfying where . Then the minimum possible value of , where with and , is _______ [2020]

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 _______ [2020]

Smart Achievers · Accepted Answer

(8)

$Let z = x + i y$

$z^{4} - | z |^{4} = 4 i z^{2}$

$\Rightarrow z^{4} - {(z \bar{z})}^{2} = 4 i z^{2} \Rightarrow z^{2} (z^{2} - {\bar{z}}^{2}) = 4 i z^{2}$

$\Rightarrow z = 0 or z^{2} - {\bar{z}}^{2} = 4 i$

$\Rightarrow 4 i x y = 4 i \Rightarrow x y = 1$

$Locus of z is a rectangular hyperbola x y = 1$

$Given that Re (z_{1}) > 0 and Re (z_{2}) < 0$

$∴$ $| z_{1} - z_{2} |_{\min} = \sqrt{{(1 + 1)}^{2} + {(1 + 1)}^{2}} = \sqrt{8}$

$\Rightarrow | z_{1} - z_{2} |_{\min}^{2} = 8$

Solution

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Solution

Q. 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 _______ [2020]

Ans. (8) Let z=x+iy z4-|z|4=4iz2 ⇒z4-(zz¯)2=4iz2⇒z2(z2-z¯ 2)=4iz2 ⇒z=0 or z2-z¯ 2=4i ⇒4ixy=4i⇒xy=1 Locus of z is a rectangular hyperbola xy=1 Given that Re(z1)>0 and Re(z2)<0 ∴ |z1-z2|min=(1+1)2+(1+1)2=8 ⇒|z1-z2|min2=8

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