Let \({z}_{1},{z}_{2}\text{and }{z}_{3}\) be three complex numbers on the circle \(|z|=1\text{ with }\arg \left({z}_{1}\…

\(\text{Given,}\\ {z}_{1}={e}^{-i\pi /4},{z}_{2}=1,{z}_{3}={e}^{i\pi /4}\\ \text{Now,}\\ {\left|{z}_{1}{\overset{¯}{z}}_{2}+{z}_{2}{\overset{¯}{z}}_{3}+{z}_{3}{\overset{¯}{z}}_{1}\right|}^{2}\\ ={\left|{e}^{-i\frac{\pi }{4}}\times 1+1\times {e}^{-i\frac{\pi }{4}}+{e}^{i\frac{\pi }{4}}\times {e}^{i\frac{\pi }{4}}\right|}^{2}\\ ={\left|{e}^{-i\frac{\pi }{4}}+{e}^{-i\frac{\pi }{4}}+{e}^{i\frac{\pi }{2}}\right|}^{2}\\ ={\left|2\left(\cos \frac{\pi }{4}-i\sin \frac{\pi }{4}\right)+\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right|}^{2}\\ =|\sqrt{2}-\sqrt{2}\mathrm{i}+\mathrm{i}{|}^{2}\\ =(\sqrt{2}{)}^{2}+(1-\sqrt{2}{)}^{2}\\ =5-2\sqrt{2}\\ \alpha =5,\beta =-2\\ \Rightarrow {\alpha }^{2}+{\beta }^{2}=29\)