If z and are two complex numbers such that |z|=1 and (z)-()=2, then z is equal to:

Question

If $z$ and $\omega$ are two complex numbers such that $|z\omega|=1$ and $\arg(z)-\arg(\omega)=\dfrac\pi2$, then $z\omega$ is equal to:

Accepted Answer

From $|z\omega|=1$ and $\arg\left(\dfrac z\omega\right)=\dfrac\pi2$: $\dfrac z\omega$ is purely imaginary, giving $(z\omega)^2=-1 \implies z\omega=\pm i$; checking the argument condition confirms $z\omega=-i.$

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