Question

If $w$ is a non-real complex number such that $\frac{w-\overline{w}z}{1-z}$ is a real number, then the set of values of $z$ is

• A. $\{z:z=\overline{z}\}$
• B. $\{z:\left|z\right|=1\}$
• C. $\{z:z\ne 1\}$
• D. $\{z:\left|z\right|=1,z\ne 1\}$

Answer 1

Answer: (D)

$\{z:\left|z\right|=1,z\ne 1\}$

Let $j=\frac{w-\overline{w}z}{1-z}$
Since, $j$ is purely real.
Therefore, $j=\overline{j}$
$\Rightarrow \frac{w-\overline{w}z}{1-z}=\frac{\overline{w}-w\overline{z}}{1-\overline{z}}$
On simplifying,
$\Rightarrow ({\left|z\right|}^2-1)(w-\overline{w})=0$
Since, $w$ is non real complex number.
Therefore, $\left|z\right|=1$
Hence, option D is correct.