Question

If $\left(1+i\right)z=\left(1-i\right)\overline{z}$, then show that $z=-i\overline{z}$.

Answer 1

$$\begin{gathered} \left(1+i\right)z=\left(1-i\right)\overline{z} \\ \Rightarrow \frac{z}{\overline{z}}=\frac{1-i}{1+i} \\ \Rightarrow \frac{z}{\overline{z}}=\frac{1-i}{1+i}\times \frac{1-i}{1-i} \\ \Rightarrow \frac{z}{\overline{z}}=\frac{1+i^2-2i}{1-i^2} \\ \Rightarrow \frac{z}{\overline{z}}=\frac{1-1-2i}{1+1}[\because i^2=-1] \\ \Rightarrow \frac{z}{\overline{z}}=\frac{-2i}{2} \\ \Rightarrow \frac{z}{\overline{z}}=-i \\ \Rightarrow z=-i\overline{z} \end{gathered}$$

Hence, $z=-i\overline{z}$.