Question

If $z$ and $w$ are complex numbers satisfying $\bar{z}+i\bar{w}=0$ and $amp\left(z w\right)=\pi ,$ then $amp\left(w\right)$ is equal to, where $amp\left(w\right)\in \left(- \pi , \pi \right]$

• A. $\frac{\pi }{4}$
• B. $\frac{- \pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{3 \pi }{4}$

Answer 1

Answer: (A)

Given $\overset{-}{z}+i\overset{-}{w}=0\Rightarrow \overset{-}{z}=-i\overset{-}{w}$ or $z=iw$ or $\frac{z}{w}=i$
$amp\left(z\right)-amp\left(w\right)=amp \, i=\frac{\pi }{2}....\left(i\right)$
also $amp\left(z w\right)=\pi $
$amp\left(z\right)+amp\left(w\right)=\pi \ldots ..\left(i i\right)$
Adding $\left(i\right)\&\left(i i\right),$ we get,
$2amp\left(z\right)=\frac{3 \pi }{2}$
$\Rightarrow amp\left(z\right)=\frac{3 \pi }{4}$
Also, $amp\left(w\right)=\frac{\pi }{4}$