Let $\bar{z}$ denote the complex conjugate of a complex number $z$ If $z$ is a non-zero complex number for which both real and imaginary parts of ${(\bar{z})}^{2} + \frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $| z | ?$

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${(\frac{43 + 3 \sqrt{205}}{2})}^{\frac{1}{4}}$

${(\frac{7 + \sqrt{33}}{4})}^{\frac{1}{4}}$

${(\frac{9 + \sqrt{65}}{4})}^{\frac{1}{4}}$

${(\frac{7 + \sqrt{13}}{6})}^{\frac{1}{4}}$

answer is A.

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Detailed Solution

Let $ω = I_{1} + i I_{2} = {(\bar{z})}^{2} + \frac{1}{z^{2}}, I_{1}, I_{2} \in I n t e g e r s$

$\Rightarrow | ω | = \sqrt{I_{1}^{2} + I_{2}^{2}} = {| z |}^{2} + \frac{1}{{| z |}^{2}} \Rightarrow {| z |}^{4} + \frac{1}{{| z |}^{4}} = I_{1}^{2} + I_{2}^{2} - 2 \Rightarrow {| z |}^{8} - (I_{1}^{2} + I_{2}^{2} - 2) {| z |}^{4} + 1 = 0 \Rightarrow {| z |}^{4} = \frac{(I_{1}^{2} + I_{2}^{2} - 2) \pm \sqrt{{(I_{1}^{2} + I_{2}^{2} - 2)}^{2} - 4}}{2} \Rightarrow | z | = {(\frac{43 \pm 3 \sqrt{205}}{2})}^{1 / 4} w h e r e I_{1} = 6, I_{2} = 3.$