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)
$\&-2xy\left(1+\frac{1}{|z{|}^4}\right)=n......$(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)
$|z{|}^4+\frac{1}{|z{|}^4}+2=\frac{7+\sqrt{13}}{6}+\frac{7-\sqrt{13}}{6}+2=\frac{14}{6}+2=\frac{7}{3}+2=\frac{13}{2}$