Question

Let $w(Im\, w \ne 0)$ 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 :

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

Answer 1

Answer: (D)

$w-\bar{w}z=k-kz $
$kz-\bar{w}z=k-w$
$z=\frac{k-w}{k-\bar{w}}\,..........\left(i\right)$
$\bar{z}=\frac{k-\bar{w}}{k-w} \,..........\left(ii\right)$
$\left(i\right) × \left(ii\right)$
$z\bar{z}=1$
$\left|z\right|=1$
but $z\ne1$