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)

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

Answer 2

The correct option is (A): $\left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}}$