Mathematics - Complex Number Question with Solution | TestHub

For $w$ to be purely imaginary, $\frac{1+\left(1-8\alpha \right)z}{1-z}+\frac{1+\left(1-8\alpha \right)\overline{z}}{1-\overline{z}}=0$

$\Rightarrow 1-\overline{z}+\left(1-8\alpha \right)z-\left(1-8\alpha \right)+1-z+\left(1-8\alpha \right)\overline{z}-\left(1-8\alpha \right)=0$

$\Rightarrow 2-\left(z+\overline{z}\right)+\left(1-8\alpha \right)\left(z+\overline{z}\right)-2+16\alpha =0$

$\Rightarrow 8\alpha \left(z+\overline{z}\right)=16\alpha$

$\Rightarrow z+\overline{z}=2$ or $\alpha =0$

For all $z\in C$, we have $\alpha =0$.