Let w (Imw0) be a complex number.

Question

Let $w$ ($\operatorname{Im}w
e0$) be a complex number. Then the set of all complex numbers $z$ satisfying $w-\bar wz=k(1-z)$, for some real number $k$, is

Accepted Answer

For $k$ real, $\dfrac{w-\bar wz}{1-z}$ must equal its own conjugate. Expanding and simplifying yields $(w-\bar w)|z|^2=w-\bar w \implies |z|^2=1$ (since $\operatorname{Im}w
e0$), and $z
e1$ (else undefined).

Solutions · 1