Let z and w be two complex numbers such that | vedclass

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(A) Given,$\bar{z}+i \bar{w}=0$. Taking the conjugate on both sides,we get $z-i w=0$,which implies $z=i w$. We are given $\operatorname{Arg}(z w)=\pi$

Vedclass · Accepted Answer

$\frac{3 \pi}{4}$

Vedclass · Answer

(A) Given,$\bar{z}+i \bar{w}=0$. Taking the conjugate on both sides,we get $z-i w=0$,which implies $z=i w$. We are given $\operatorname{Arg}(z w)=\pi$. Using the property $\operatorname{Arg}(z w) = \operatorname{Arg}(z) + \operatorname{Arg}(w)$,we have $\operatorname{Arg}(z) + \operatorname{Arg}(w) = \pi$. Since $z=i w$,we have $w = \frac{z}{i} = -iz$. Thus,$\operatorname{Arg}(w) = \operatorname{Arg}(-i) + \operatorname{Arg}(z) = -\frac{\pi}{2} + \operatorname{Arg}(z)$. Substituting this into the equation: $\operatorname{Arg}(z) + (\operatorname{Arg}(z) - \frac{\pi}{2}) = \pi$. $2 \operatorname{Arg}(z) = \pi + \frac{\pi}{2} = \frac{3 \pi}{2}$. Therefore,$\operatorname{Arg}(z) = \frac{3 \pi}{4}$.

Let $z$ and $w$ be two complex numbers such that $\bar{z}+i \bar{w}=0$ and $\operatorname{Arg}(z w)=\pi$. Then,$\operatorname{Arg} z=$

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