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Polar Representation of a Complex Number

Let z1,z2,z3...

Question

Let $z_{1}, z_{2}, z_{3}$ be three distinct complex numbers lying on a circle whose centre is at the origin. If $z_{i} + z_{j} z_{k},$ where $i, j, k \in {1, 2, 3}$ and $i \neq j \neq k$ are real numbers, then the value of $4 (z_{1} \times z_{2} \times z_{3})$ is

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Solution

As $z_{1}, z_{2}, z_{3}$ lie on the circle centered at origin.
$∴ | z_{1} | = | z_{2} | = | z_{3} | = r$
(where $r$ is the radius of the circle)
Let $z_{1} = r e^{i θ_{1}}$
$z_{2} = r e^{i θ_{2}}$
$z_{3} = r e^{i θ_{3}}$
It is given that $z_{1} + z_{2} z_{3} \in R$
$∴ z_{1} = ¯ ¯¯¯¯¯¯¯ ¯ z_{2} z_{3} \Rightarrow r e^{i θ_{1}} = r e^{- i θ_{2}} \cdot r e^{- i θ_{3}} \Rightarrow r = r^{2} e^{- i (θ_{1} + θ_{2} + θ_{3})} \Rightarrow r e^{- i (θ_{1} + θ_{2} + θ_{3})} = 1 (∵ r \neq 0)$
Taking mod both sides, we get
$r = 1$

Hence, $4 (z_{1} \times z_{2} \times z_{3})$
$= 4 (¯ ¯¯¯¯¯¯¯ ¯ z_{2} z_{3} \times z_{2} z_{3}) = 4 (| z_{2} | \times | z_{3} |)^{2} = 4 (∵ | z_{2} | = | z_{3} | = 1)$

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