The set of all R, for which w=1+(1-8)z1-z is a purely imaginary number, for all z C satisf…

Question

The set of all $\alpha\in\mathbb R$, for which $w=\dfrac{1+(1-8\alpha)z}{1-z}$ is a purely imaginary number, for all $z\in\mathbb C$ satisfying $|z|=1$ and $\operatorname{Re}(z)
e1$, is

Accepted Answer

With $z=x+iy$, $x^2+y^2=1$, setting $\operatorname{Re}(w)=0$ for all such $z$ forces $1-8\alpha=1 \implies \alpha=0$. So the set is $\{0\}$.

Solutions · 1