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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 $|z|$ ?
$\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$
$\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$
$\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$
$\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$
Solution
Let $(\bar{z})^2+\frac{1}{z^2}=m+$ in, $m, n \in Z$
$(\bar{z})^2+\frac{\bar{z}^2}{|z|^2}= m + in$
$\Rightarrow\left( x ^2- y ^2\right)\left(1+\frac{1}{|z|^4}\right)= m$ $. . . . (1)$
$\&-2 xy \left(1+\frac{1}{|z|^2}\right)= n$ $. . . . (2)$
Equation $(1)^2+(2)^2$
$\left(1+\frac{1}{|z|^2}\right)^2\left[\left(x^2+y^2\right)^2\right]=m^2+n^2$
$\left(1+\frac{1}{|z|^2}\right)^2(|z|)^2=m^2+n^2$
$\Rightarrow|z|^2+\frac{1}{|z|^2}+2=m^2+ n ^2$
Now for option $(A)$
$|z|^4=\frac{43+3 \sqrt{205}}{2}$
$\Rightarrow m^2+ n ^2=45$
$\Rightarrow m = \pm 6, n = \pm 3$
Option $(B)$
$|z|^2+\frac{1}{|z|^2}+2=\frac{7+\sqrt{33}}{4}+\frac{7-\sqrt{33}}{4}+2=\frac{7}{2}+2=\frac{11}{2}$
Option $(C)$
$|z|^2+\frac{1}{|z|^2}+2=\frac{9+\sqrt{65}}{4}+\frac{9-\sqrt{65}}{4}+2=\frac{18}{4}+2=\frac{9}{2}+2=\frac{13}{2}$
Option $(D)$
$|z|^4+\frac{1}{|z|^2}+2=\frac{7+\sqrt{13}}{6}+\frac{7-\sqrt{13}}{6}+2=\frac{14}{6}+2=\frac{7}{3}+2=\frac{13}{2}$
