​ Let z denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary

Question

Let \(\overline{z}\) 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 \((\overline{z})^2 +\frac{1}{z^2}\) are integers, than which of the following is/are possible value(s) of |z|?

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

(B) \((\frac{7+\sqrt{33}}{4})^{1/4}\)

(C) \((\frac{9+\sqrt{65}}{4})^{1/4}\)

(D) \((\frac{7+\sqrt{13}}{6})^{1/4}\)

Answer 1

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