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Letw\left ( Im\: w\neq 0 \right ) be a complex number. Then the set of all complex numbers z satisfying the equation w-\bar{w}z=k\left ( 1-z \right ) for some real number k, is :

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\\\omega -\bar{\omega }z=k(1-z)\\ \omega -k=z(\bar{\omega }-k)\\ z=\frac{\omega -k}{\bar{\omega }-k}\\ z=\frac{\omega -k}{\bar{\omega }-\bar{k}}

z is of the form \frac{C}{\bar{C}}

Therefore |z| = 1\ and\ z\neq 1

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