Let $\mathrm{w}(\mathrm{Im} \mathrm{w}\ne 0)$ be a complex number. Then the set of all complex numbers $\mathrm{z}$ satisfing the equation $\mathrm{w}-\overline{\mathrm{wz}}=\mathrm{k}(1-\mathrm{z}),$ for some real number $\mathrm{k},$ is

$\text{ let ' }\mathrm{w}\text{ ' be }\mathrm{a}+\mathrm{ib}\text{ and }\mathrm{z}\text{ be ' }\mathrm{x}+\mathrm{iy}\text{ ' }$

$\begin{array}{r}a+ib-(a-ib)(x+iy)=k(1-x-iy)\\ a+ib-[ax+ayi-bxi+by]=k(1-x)-iky\\ a+ib-ax-ayi+bxi-by=k(1-x)-kyi\\ (a-ax-by)+i(b-ay+bx)=k(1-x)-kyi\end{array}$

$\begin{array}{l}k=\frac{(a-ax-by)}{(1-x)},k=\left(\frac{ay-bx-b}{y}\right)\\ \frac{a-ax-by}{1-x}=\frac{ay-bx-b}{y}\\ ay-axy-b{y}^2=ay-bx-b-ayx+b{x}^2+bx\\ -b{y}^2=b{x}^2-b\\ b=b{y}^2+b{x}^2\\ x^2+y^2=1\\ \left|z\right|=1\end{array}$