Let denote the complex conjugate of a complex number . If is a non-zero complex number for which both real and imaginary parts of are integers, then which of the following is/are possible value(s) of ? [2022]

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 $| z |$ ? [2022]

Smart Achievers · Accepted Answer

(1)

$Let z = r . e^{i θ} \Rightarrow \bar{z} = r e^{- i θ}$

$∴$ ${(\bar{z})}^{2} + \frac{1}{z^{2}} = r^{2} e^{- 2 i θ} + \frac{1}{r^{2} e^{2 i θ}} = (r^{2} + \frac{1}{r^{2}}) e^{- 2 i θ} = a + i b (say),$

$where a, b \in ℤ$

$So, {(r^{2} + \frac{1}{r^{2}})}^{2} = a^{2} + b^{2}$ $\Rightarrow r^{8} - (a^{2} + b^{2} - 2) r^{4} + 1 = 0$

$\Rightarrow r^{4} = \frac{(a^{2} + b^{2} - 2) \pm \sqrt{{(a^{2} + b^{2} - 2)}^{2} - 4}}{2}$

$For option (a): | z |^{4} = \frac{43 + 3 \sqrt{205}}{2}$

$\Rightarrow a^{2} + b^{2} = 45 i.e. (a, b) = (\pm 6, \pm 3) or (\pm 3, \pm 6)$

$For option (b): | z |^{4} = \frac{7 + \sqrt{33}}{4} \Rightarrow a^{2} + b^{2} = \frac{11}{2}$

$For option (c): a^{2} + b^{2} = \frac{13}{2}$

$For option (d): a^{2} + b^{2} = \frac{13}{3}$

Solution

Q.
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 $| z |$ [2022]

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

2 ${(\frac{7 + \sqrt{33}}{4})}^{\frac{1}{4}}$

3 ${(\frac{9 + \sqrt{65}}{4})}^{\frac{1}{4}}$

4 ${(\frac{7 + \sqrt{13}}{6})}^{\frac{1}{4}}$

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Solution

Q. 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 parts of (z¯)2+1z2 are integers, then which of the following is/are possible value(s) of |z| [2022]

1 (43+32052)14

2 (7+334)14

3 (9+654)14