Question

Let ¯z denote the complex conjugate of a complex number ?. If ? is a non-zero complex number for which both real and imaginary parts of (¯z)²+1/z²

are integers, then which of the following is/are possible value(s) of |z| ?
A. ((43+3√205)/2)¹/⁴
B. ((7+√3)/4)¹/⁴
C. ((9+√65)/4)¹/⁴
D. ((7+√13)/6)¹/⁴

Answer 1

The possible values for |z|, where z is a non-zero complex number satisfying the conditions, are given by A. ((43 + 3√205)/2)¹/⁴. The conditions are derived from the requirement that both real and imaginary parts of (¯z)² + 1/z² are integers.

Let's denote z = a + bi, where a and b are the real and imaginary parts of z, respectively.

The complex conjugate of z is
`\( \bar{z} = a - bi \)`.

Now, we can express
`\( \bar{z}^2 + (1)/(z^2) \)` and simplify to find conditions on a and b such that both the real and imaginary parts are integers.

`\[ \bar{z}^2 + (1)/(z^2) = (a - bi)^2 + (1)/((a + bi)^2) \]\[ = (a^2 - 2abi - b^2) + (1)/(a^2 + 2abi - b^2) \]`

Now, equate the real and imaginary parts to integers:

1. For the real part to be an integer:

`\[ a^2 - b^2 + (1)/(a^2 - b^2) \]\[ = ((a^2 - b^2)^2 + 1)/(a^2 - b^2) \]`

The numerator must be divisible by the denominator, so
`(a ^ 2 - b ^ 2) ^ 2 + 1`must be a multiple of
`\( a^2 - b^2 \)`. This implies
`\( a^2 - b^2 = 1 \)`.

2. For the imaginary part to be an integer:

`\[ -2ab - (2ab)/(a^2 + b^2) \]\[ = (-2ab(a^2 + b^2) - 2ab)/(a^2 + b^2) \]`

This implies a^2 + b^2 must divide 2ab, and since a^2 + b^2 = 1, either a = 0 or b = 0.

Considering both conditions, we have
`\( a^2 - b^2 = 1 \)` and either a = 0 or b = 0. These conditions are satisfied by the points on the unit circle where
`\( a = \cos(\theta) \) and \( b = \sin(\theta) \)`.

Therefore, the possible values for |z| are
`\( 1^(1/4) \)`, and the correct option is A. ((43 + 3√205)/2)¹/⁴.