Question

If $z$ and $w$ are two non-zero complex numbers such that $\left|z w\right|=1$ and $arg\left(z\right)-arg\left(w\right)=\frac{\pi }{2}$ , then the value of $5i\overset{-}{z}w$ is equal to

• A. $-5$
• B. $5i$
• C. $5$
• D. $-5i$

Answer 1

Answer: (C)

$arg\left(\frac{z}{w}\right)=\frac{\pi }{2}\Rightarrow \frac{z}{w}=\left|\frac{z}{w}\right|e^{\left(i \pi \right)/2}$
$\Rightarrow \frac{z}{w}=\left|\frac{z}{w}\right|i$
$\Rightarrow w=z\frac{\left|w\right|}{\left|z\right|}\left(- i\right)$
$\Rightarrow w\overset{-}{z}=\frac{\overset{-}{z} z \left|w\right|}{\left|z\right|}\left(- i\right)$
$\Rightarrow 5i\overset{-}{z}w=5\left(- i^{2}\right)\frac{\left(\left|z\right|\right)^{2} \left|w\right|}{\left|z\right|}$
$=5\left(1\right)\left|z\right|\left|w\right|$
$=5$