Let |Zi| = 1 for i = 1, 2, 3 satisfying |z1z2 + z2z3 + z3z1|^2 = a+b√2, where a, b an rational numbers such that arg(z1) = -π/4, then find (a, b)

Question

 Let \(\left|Z_i\right|=1 \text{for i}=1,2,3\) satisfying \(\left|\overline{Z}_1 z_2+\overline{Z}_2 z_3+\overline{Z}_3 z_1\right|^2=a+b \sqrt{2},\) where a, b an rational numbers such that \(\arg \left(Z_1\right)=\frac{\pi}{4}, \arg \left(Z_2\right)=0\ \text{ and} \arg \left(Z_3\right)=\frac{-\pi}{4},\) then find (a, b)

(1) (5, 2)

(2) (-5, -2)

(3) (5, -2)

(4) (-5, 2)

Answer 1

Correct option is (3) (5, -2)

\(Z_1=|1| e^{i \frac{\pi}{4}}=\frac{1}{\sqrt{2}}+i \cdot \frac{1}{\sqrt{2}}\)  

\(Z_2=|1| e^{-(0)} 1+0 i\)   

\(Z_3=|1| e^{-i \frac{\pi}{4}}=\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\) 

\(\bar{Z}_1 z_2=\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)(1)\)