If (1+i)z=(1-i) z , then show that z=-i z . - Vidyadip

Question

If $(1+\text{i})\text{z}=(1-\text{i})\bar{\text{z}},$ then show that $\text{z}=-\text{i}\bar{\text{z}}.$

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Answer

$(1+\text{i})\text{z}=(1-\text{i})\bar{\text{z}}$ $\Rightarrow\text{z}=\frac{(1-\text{i})}{(1+\text{i})}\bar{\text{z}}$ $\Rightarrow\text{z}=\frac{(1-\text{i})(1-\text{i})}{(1+\text{i})(1-\text{i})}\bar{\text{z}}$ [Rationalizing the denominator] $\Rightarrow\text{z}=\frac{(1-2\text{i}-1)}{(1+1)}\bar{\text{z}}$ $\Rightarrow\text{z}=\frac{-2\text{i}}{2}\bar{\text{z}}$ $\Rightarrow\text{z}=-\text{i}\bar{\text{z}}$

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