For a complex number z, let Rez 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-z22, where z1,z2∈S with Rez1>0 and Rez2<0, is _______

JEE Advanced · Mathematics · 3. 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 $i = \sqrt{- 1}$ . 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 _______

Verified Solution

Answer & Solution

Correct Answer

(D) 8

Step-by-step Solution

Detailed explanation

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^{2} - {\bar{z}}^{2} = 4 i

\Rightarrow (z + \bar{z}) (z - \bar{z}) = 4 i

\Rightarrow (\frac{z + \bar{z}}{2}) (\frac{z - \bar{z}}{2 i}) = 1 \Rightarrow x y = 1

for

z_{1} & z_{2} \Rightarrow x_{1} y_{1} = 1

and

x_{2} y_{2} = 1

x_{1} & x_{2}

are of opposite sign, similarly

y_{1}

y_{2}

are of opposite sign

\Rightarrow x_{1} > 0, y_{1} > 0, x_{2} < 0, y_{2} < 0

Now

{|z_{1} - z_{2}|}^{2} = {(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}

= x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2} - 2 x_{1} x_{2} - 2 y_{1} y_{2}

$=x_1^2+x_2^2+y_1^2+y_2^2+x_1\left(-x_2\right)+x_1$ $\left(-x_2\right)+y_1\left(-y_2\right)+y_1\left(-y_2\right) $
$ \geq 8(x_1^2 \cdot x_2^2 \cdot y_1^2 \cdot y_2^2 \cdot x_1\left(-x_2\right) \cdot x_1\left(-x_2\right) \cdot$ $y_1\left(-y_2\right) \cdot y_1\left(-y_2\right)$$^{1 / 8}$

\geq 8 {({(x_{1} y_{1})}^{4} {(x_{2} y_{2})}^{4})}^{1 / 8}

\geq 8

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Column I	Column II
(A) In $R ^2$, if the magnitude of the projection vector of the vector $\alpha \hat{ i }+\beta \hat{ j }$ on $\sqrt{3} \hat{ i }+\hat{ j }$ is $\sqrt{3}$ and if $\alpha=2+\sqrt{3} \beta$, then possible value (s) of $\|\alpha\|$ is (are)	(P) 1
(B) Let a and b be real numbers such that the function $f ( x )=\left\{\begin{array}{cc}-3 ax ^2-2, & x <1 \\ bx + a ^2, & x \geq 1\end{array}\right.$ is differentiable for all $x \in R$. Then possible value ( s ) of a is (are)	(Q) 2
(C)Let $\omega \neq 1$ be a complex cube roots of unity. If $\left(3-3 \omega+2 \omega^2\right)^{4 n+3}$ $+~\left(2+3 \omega-3 \omega^2\right)^{4 n+3}$ $~+\left(-3+2 \omega+3 \omega^2\right)^{4 n+3}=0$ then possible value $(s)$ of $n$ is (are)	(R) 3
(D) Let the harmonic mean of two positive real numbers $a$ and $b 4$. If $q$ is a positive real number such that $a , 5, q , b$ is an arithmetic progression, then the value $( s )$ of $\| q - a \|$ is (are)	(S) 4
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Column I	Column II
(A) In \(R ^2\), if the magnitude of the projection vector of the vector \(\alpha \hat{ i }+\beta \hat{ j }\) on \(\sqrt{3} \hat{ i }+\hat{ j }\) is \(\sqrt{3}\) and if \(\alpha=2+\sqrt{3} \beta\), then possible value (s) of \(\|\alpha\|\) is (are)	(P) 1
(B) Let a and b be real numbers such that the function \(f ( x )=\left\{\begin{array}{cc}-3 ax ^2-2, & x <1 \\ bx + a ^2, & x \geq 1\end{array}\right.\) is differentiable for all \(x \in R\). Then possible value ( s ) of a is (are)	(Q) 2
(C)Let \(\omega \neq 1\) be a complex cube roots of unity. If \(\left(3-3 \omega+2 \omega^2\right)^{4 n+3}\) \(+~\left(2+3 \omega-3 \omega^2\right)^{4 n+3}\) \(~+\left(-3+2 \omega+3 \omega^2\right)^{4 n+3}=0\) then possible value \((s)\) of \(n\) is (are)	(R) 3
(D) Let the harmonic mean of two positive real numbers \(a\) and \(b 4\). If \(q\) is a positive real number such that \(a , 5, q , b\) is an arithmetic progression, then the value \(( s )\) of \(\| q - a \|\) is (are)	(S) 4
	(T) 5