JEE Mains · Maths · STD 11 - 4.1 complex nubers
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 :
- A \(24\)
- B \(29\)
- C \(41\)
- D \(31\)
Answer & Solution
Correct Answer
(B) \(29\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & |z|=1 \\ & \arg \left(z_1\right)=-\frac{\pi}{4}, \arg \left(z_2\right)=0, \arg \left(z_3\right)=\frac{\pi}{4} \\ & z_1=|1| e^{-\frac{\pi}{4}}=\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…
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