Mathematics - Complex Number Question with Solution | TestHub

Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_1\right)=\frac{-\pi }{4},\arg \left(z_2\right)=0$ and $\arg \left(z_3\right)=\frac{\pi }{4}$. If ${\left|z_1{\bar{z}}_2+z_2{\bar{z}}_3+z_3{\bar{z}}_1\right|}^2=\alpha +\beta \sqrt{2},\alpha ,\beta \in \mathbf{Z}$, then the value of ${\alpha }^2+{\beta }^2$ is :