Two complex numbers z_1 and z_2 satisfy |z_2|=1 and dfrac z_1-2z_2 2-z_1bar z _2 is unimodular, but z_2 is not unimodular. Where does z_1 lie?

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EEJ MAIN Mathematics QUESTION #6036

Question 1

Two complex numbers $z_1$ and $z_2$ satisfy $|z_2|=1$ and $\dfrac{z_1-2z_2}{2-z_1\bar{z}_2}$ is unimodular, but $z_2$ is not unimodular. Where does $z_1$ lie?

On a line parallel to the $x$-axis
On a line parallel to the $y$-axis
On a circle of radius 2✔️
On a circle of radius $\sqrt{2}$

Correct Answer Explanation

Let $w = \frac{z_1-2z_2}{2-z_1\bar{z}_2}$, with $|w|=1$.

$|z_1-2z_2|^2 = |2-z_1\bar{z}_2|^2$

$(z_1-2z_2)\overline{(z_1-2z_2)} = (2-z_1\bar{z}_2)\overline{(2-z_1\bar{z}_2)}$

$|z_1|^2 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + 4|z_2|^2 = 4 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + |z_1|^2|z_2|^2$

$|z_1|^2 + 4|z_2|^2 = 4 + |z_1|^2|z_2|^2$

$|z_1|^2(1-|z_2|^2) = 4(1-|z_2|^2)$

Since $|z_2|\neq1$: $|z_1|^2=4\Rightarrow|z_1|=2$. So $z_1$ lies on a circle of radius 2.

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