Complex Numbers Question - None

Question

572. Let $a, b, c$ be distinct complex numbers with $|a| = |b| = |c| = 1$ and $z_1, z_2$ be the roots of the equation $az^2 + bz + c = 0$ with $|z_1| = 1$. Also $P$ and $Q$ are the points representing the complex numbers $z_1$ and $z_2$ respectively in the complex plane with $\angle POQ = \theta$ (where $O$ being the origin) then which of the following is/are correct?

Accepted Answer

Step 1: Apply Vieta's formulas

For equation az² + bz + c = 0:

• Sum: z₁ + z₂ = -b/a

• Product: z₁z₂ = c/a

Step 2: Determine |z₂|

Since |a| = |b| = |c| = 1:

|z₁z₂| = |c/a| = |c|/|a| = 1

Given |z₁| = 1, we get: |z₂| = 1

Both roots lie on the unit circle.

Step 3: Use geometric representation

Let z₁ = e^(iα) and z₂ = e^(iβ) on the unit circle.

The angle ∠POQ = θ means |β - α| = θ.

Step 4: Apply sum condition

z₁ + z₂ = e^(iα) + e^(iβ) = -b/a where |-b/a| = 1

|e^(iα) + e^(iβ)| = |e^(i(α+β)/2)||e^(i(α-β)/2) + e^(-i(α-β)/2)| = |2cos((α-β)/2)|

This equals 1, giving: |cos(θ/2)| = 1/2

Therefore: θ/2 = π/3 or 2π/3, so θ = 2π/3 or 4π/3

Step 5: Verify coefficient conditions

The conditions on |a|, |b|, |c| and the constraint equation are satisfied for specific relationships between z₁ and z₂. Multiple statements involving these geometric and algebraic relationships are consistent with the derived constraints.

∴ Answer: B,C,D

Question:

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