Let $Z$ and $W$ be complex numbers such that $|Z|=|W|$, and $\arg Z$ denotes the principal argument of $Z$. Statement 1:If $\arg Z+\arg W=\pi$, then $Z=-\bar{W}$. Statement 2: $|Z|=|W|$, implies arg $Z-\arg \bar{W}=\pi$.

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Aicharya · Accepted Answer

Let $|Z|=|\mathrm{W}|=r$ $ \Rightarrow Z=r e^{i 	heta}, \mathrm{W}=r e^{i \phi} $ where $	heta+\phi=\pi$ $ 	herefore \quad \bar{W}=r e^{-i \phi} $ Now, $Z=r e^{i(\pi-\phi)}=r e^{i \pi} 	imes e^{-i \phi}=-r e^{-i \phi}$ $ =-\bar{W} $ Thus, statement-1 is true but statement- 2 is false

Let $Z$ and $W$ be complex numbers such that $|Z|=|W|$, and $\arg Z$ denotes the principal argument of $Z$.…

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