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(C) Given $\overline{z}_{1} = i \overline{z}_{2}$. Taking conjugate on both sides,we get $z_{1} = -i z_{2}$.Substitute $z_{1}$ into the argument equat

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$\arg z_{1} = \frac{\pi}{4}$

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(C) Given $\overline{z}_{1} = i \overline{z}_{2}$. Taking conjugate on both sides,we get $z_{1} = -i z_{2}$.Substitute $z_{1}$ into the argument equation: $\arg \left( \frac{-i z_{2}}{\overline{z}_{2}} ight) = \pi$.Using properties of arguments,$\arg(-i) + \arg \left( \frac{z_{2}}{\overline{z}_{2}} ight) = \pi$.We know $\arg(-i) = -\frac{\pi}{2}$ and $\arg \left( \frac{z_{2}}{\overline{z}_{2}} ight) = \arg(z_{2}) - \arg(\overline{z}_{2}) = 	heta - (-	heta) = 2	heta$,where $	heta = \arg(z_{2})$.So,$-\frac{\pi}{2} + 2	heta = \pi$,which gives $2	heta = \frac{3\pi}{2}$,so $	heta = \frac{3\pi}{4}$.Now,$z_{1} = -i z_{2} = e^{-i\pi/2} \cdot |z_{2}| e^{i(3\pi/4)} = |z_{2}| e^{i(3\pi/4 - \pi/2)} = |z_{2}| e^{i\pi/4}$.Thus,$\arg(z_{1}) = \frac{\pi}{4}$.

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overline{z}_{1} = i \overline{z}_{2}$ and $\arg \left( \frac{z_{1}}{\overline{z}_{2}} \right) = \pi$. Then:

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