Question

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 $|z|$ ?

• A. $\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$
• B. $\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$
• C. $\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$
• D. $\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$

Answer 1

Answer: (A)

Let $(\overline{ z })^2+\frac{1}{ z ^2}= m + in , m , n \in Z$
$(\overline{ z })^2+\frac{\overline{ z }^2}{| z |^4}= 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 |^4}\right)= n......$(2)
Equation $(1)^2+(2)^2$
$ \left(1+\frac{1}{|z|^4}\right)^2\left[\left( x ^2+ y ^2\right)^2\right]= m ^2+ n ^2 $
$ \left(1+\frac{1}{| z |^4}\right)^2(| z |)^4= m ^2+ n ^2 $
$\Rightarrow| z |^4+\frac{1}{| z |^4}+2= m ^2+ n ^2$
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)
$|z|^4+\frac{1}{|z|^4}+2=\frac{7+\sqrt{33}}{4}+\frac{7-\sqrt{33}}{4}+2=\frac{7}{2}+2=\frac{11}{2}$
Option (C)
$|z|^4+\frac{1}{|z|^4}+2=\frac{9+\sqrt{65}}{4}+\frac{9-\sqrt{65}}{4}+2=\frac{18}{4}+2=\frac{9}{2}+2=\frac{13}{2}$
Option (D)
$|z|^4+\frac{1}{|z|^4}+2=\frac{7+\sqrt{13}}{6}+\frac{7-\sqrt{13}}{6}+2=\frac{14}{6}+2=\frac{7}{3}+2=\frac{13}{2}$