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Let $w$ $(Im\, w \neq 0)$ be a complex number. Then the set of all complex number $z$ satisfying the equation $w - \overline {w}z = k\left( {1 - z} \right)$ , for some real number $k$, is
$\left\{ {z:\left| z \right| = 1} \right\}$
$\left\{ {z:z = \overline z } \right\}$
$\left\{ {z:z \ne 1} \right\}$
$\left\{ {z:\left| z \right| = 1,z \ne 1} \right\}$
Solution
Consider the equation
$w-\bar{w} z=k(1-z), k \in R$
Clearly $z \neq 1$ and $\frac{w-\bar{w} z}{1-z}$ is purely real
$\therefore \frac{\overline{w-\bar{w} z}}{1-z}=\frac{w-\bar{w} z}{1-z}$
$\Rightarrow \frac{\bar{w}-w \bar{z}}{1-\bar{z}}=\frac{w-\bar{w} z}{1-z}$
$\Rightarrow \bar{w}-\bar{w} z-w \bar{z}+w \overline{z z}$
$=w-w \bar{z}-\bar{w} z+\bar{w} z \bar{z}$
$\Rightarrow \bar{w}+w|z|^{2}=w+\bar{w}|z|^{2}$
$ \Rightarrow (w – \bar w)(|z{|^2}) = w – \bar w$
$\Rightarrow|z|^{2}=1$ $ \quad(\because \operatorname{Im} w \neq 0)$
$\Rightarrow|z|=1$ and $z \neq 1$
$\therefore$ The required set is $\{z:|z|=1, z \neq 1\}$
