Complex Numbers Question - None

Question

Let z₁ and z₂ be two roots of the equation z² + az + b = 0, z being complex number. Further, assume that the origin, z₁ and z₂ form an equilateral triangle, then:

Accepted Answer

Step 1: Set up the equilateral triangle condition.

For origin O, z₁, and z₂ to form an equilateral triangle, we need: |z₁| = |z₂| = |z₁ - z₂|

Step 2: Express z₂ in terms of z₁.

Since |z₁| = |z₂|, let |z₁| = |z₂| = r. If the triangle is equilateral with one vertex at origin, then z₂ must be z₁ rotated by ±60°:

z₂ = z₁·e^(iπ/3) or z₂ = z₁·e^(-iπ/3)

Taking z₂ = z₁·e^(iπ/3) = z₁(cos 60° + i sin 60°) = z₁(1/2 + i√3/2)

Step 3: Apply Vieta's formulas.

From z² + az + b = 0:

• Sum of roots: z₁ + z₂ = -a

• Product of roots: z₁·z₂ = b

Step 4: Calculate z₁ + z₂.

z₁ + z₂ = z₁ + z₁·e^(iπ/3) = z₁(1 + e^(iπ/3))

= z₁(1 + 1/2 + i√3/2) = z₁(3/2 + i√3/2)

Therefore: -a = z₁(3/2 + i√3/2)

Step 5: Calculate z₁·z₂.

z₁·z₂ = z₁·z₁·e^(iπ/3) = |z₁|²·e^(iπ/3)

Therefore: b = |z₁|²·e^(iπ/3)

Step 6: Find |a|².

|a|² = |z₁(3/2 + i√3/2)|² = |z₁|²·|(3/2 + i√3/2)|²

|(3/2 + i√3/2)|² = (3/2)² + (√3/2)² = 9/4 + 3/4 = 12/4 = 3

So: a² = 3|z₁|²

Step 7: Find relation between a² and b.

Since b = |z₁|²·e^(iπ/3), we have |b|² = |z₁|⁴

But more directly: |z₁|² appears in both expressions.

From a² = 3|z₁|² and |z₁|² = |b| (taking magnitude), we get:

a² = 3b

∴ Answer: C

Question:

Step-by-Step Solution

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