Let and be three complex number on the circle |z| = 1 with , and . If , then the value of is : [2025]

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Solution

Q.
Let $z_{1}, z_{2}$ and $z_{3}$ be three complex number on the circle |z| = 1 with $\arg (z_{1}) = \frac{- π}{4}$ , $\arg (z_{2}) = 0$ and $\arg (z_{3}) = \frac{π}{4}$ . If $| z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} |^{2} = α + β \sqrt{2}, α, β \in Z$ , then the value of $α^{2} + β^{2}$ is : [2025]

1 24

2 41

3 29

4 31

Ans.
(3)

Given, |z| = 1

Now, $z_{1} = e^{i \frac{π}{4}} = \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} = \frac{1 - i}{\sqrt{2}}; z_{2} = e^{i 0} = 1$ ;

$z_{3} = e^{i \frac{π}{4}} = \frac{1 + i}{\sqrt{2}}$

Now, $| z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1} |^{2}$ $=$ $| \frac{1 - i}{\sqrt{2}} \cdot 1 + 1 \cdot \frac{1 - i}{\sqrt{2}} + {(\frac{1 + i}{2})}^{2} |^{2}$

$=$ $| \sqrt{2} + (1 - \sqrt{2}) i |^{2}$ $= 5 - 2 \sqrt{2} = α + β \sqrt{2}$

$\Rightarrow α = 5 and β = - 2$ $∴ α^{2} + β^{2} = 29$

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