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 $\left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1\right|^2=\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}$,…

Question

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 :

Aicharya · Accepted Answer

$\begin{aligned} & |z|=1 \ & \arg \left(z_1ight)=-\frac{\pi}{4}, \arg \left(z_2ight)=0, \arg \left(z_3ight)=\frac{\pi}{4} \ & z_1= \frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}} \ & z_2=1+0 i \ & z_3=\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}} \ & z_1 \bar{z}_2=\frac{1-i}{\sqrt{2}} \ & z_2 \bar{z}_3=\frac{1-i}{\sqrt{2}} \ & z_3 \bar{z}_1=\frac{(1+i)^2}{2} \ & z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1=\sqrt{2}+i(1-\sqrt{2}) \ & \left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1ight|^2=5-2 \sqrt{2} \ & \alpha=5, \beta=-2 \ & \alpha_2^2+\beta^2=29\end{aligned}$

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg…

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