Complex Numbers Part 1

Given that $|z \omega|=1$ $\Rightarrow|z||\omega|=1$ $\Rightarrow|z|=\;\frac{1}{|\omega|}$ and $\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\;\frac{\pi}{2}$ $\Rightarrow \operatorname{Arg} \left(\;\frac{z}{\omega}\right) =\;\frac{\pi}{2}$ When argument of a complex number is $\;\frac{\pi}{2}$, it means it is making an angle of $\;\frac{\pi}{2}$ with the real axis in the counterclockwise, so it is along the imaginary axis and positive side of imaginary axis. So, $\;\frac{z}{\omega}$ is a purely imaginary number that means there is no real part in this complex number. So we can assume, $\;\frac{z}{\omega}=k i$ $\Rightarrow \left|\;\frac{z}{\omega}\right| = |k i|$ $\Rightarrow \left|\;\frac{z}{\omega}\right| = |k||i|$ $\Rightarrow \left|\;\frac{z}{\omega}\right| = k \;\;$ [as $|i|=1]$ $\Rightarrow|z| \times \;\frac{1}{|\omega|}=k$ $\Rightarrow|z| \times|z|=k \;\; [.$ as $\;\frac{1}{|\omega|}=|z|]$ $\Rightarrow|z|^2=k$ $\Rightarrow|z|=\sqrt{k}$ $\therefore |\omega|=\;\frac{1}{\sqrt{k}}$ As $\;\frac{z}{\omega}$ is imaginary so we can write, $\;\frac{z}{\omega} = -\;\frac{\overline{z}}{\overline{\omega}}$ [When $z$ is imaginary then $z=-\overline{z}$ ] $\Rightarrow \overline{z} \omega = -z \overline{\omega}$$\Rightarrow \overline{z} \omega = -\;\frac{z}{\omega} \cdot \overline{\omega} \cdot \omega$$\Rightarrow \overline{z} \omega = -\;\frac{z}{\omega} \cdot |\omega|^2$$\Rightarrow \overline{z} \omega = -k i \cdot \left(\;\frac{1}{\sqrt{k}}\right)^2$$\Rightarrow \overline{z} \omega = -k i \cdot \;\frac{1}{k}$$\Rightarrow \overline{z} \omega = -i$