WBJEE · Maths · Complex Number

Let \(z_{1}\) be a fixed point on the circle of radius 1 centred at the origin in the argand plane and \(z_{1} \neq \pm 1 .\) Consider an equilateral triangle inscribed in the circle with \(z_{1}, z_{2}, z_{3}\) as the
direction. Then, \(z_{1} z_{2} z_{3}\) is equal to

A \(z_{1}^{2}\)
B \(z_{1}^{3}\)
C \(z_{1}^{4}\)
D \(z_{1}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(z_{1}^{3}\)

Step-by-step Solution

Detailed explanation

Given, \(z_{1} \neq\pm 1\) Since, \(z_{2}\) and \(z_{3}\) can be obtained by rotating vector representing through \(\frac{2 \pi}{3}\) and \(\frac{4 \pi}{3}\) respectively \(\therefore \quad z_{2}=z_{1} \omega\) and \[ z_{3}=z_{1} \omega^{2} \]…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let \(z_{1}\) be a fixed point on the circle of radius 1 centred at the origin in the argand plane and \(z_{1} \neq \pm 1 .\) Consider an equilateral triangle inscribed in the circle with \(z_{1}, z_{2}, z_{3}\) as the direction. Then, \(z_{1} z_{2} z_{3}\) is equal to

Step-by-step Solution

Let \(z_{1}\) be a fixed point on the circle of radius 1 centred at the origin in the argand plane and \(z_{1} \neq \pm 1 .\) Consider an equilateral triangle inscribed in the circle with \(z_{1}, z_{2}, z_{3}\) as the
direction. Then, \(z_{1} z_{2} z_{3}\) is equal to