<p>Let \(z_1\) and \(z_2\) be two roots of the equation \(z^2 + az + b = 0\), \(z\) being complex. Further, assume that the origin, \(z_1\) and \(z_2\) form an equilateral triangle, then</p>
Step-by-Step Solution
Key Concept: For an equilateral triangle with one vertex at origin, the two complex roots must satisfy |z₁| = |z₂| = |z₁ - z₂|, which means z₂ = z₁e^(±iπ/3). Combined with Vieta's formulas, this gives a direct relationship between coefficients a and b.
<p><strong>Step 1:</strong> For equilateral triangle with vertices at 0, z₁, z₂: we need |z₁| = |z₂| = |z₁ - z₂|</p><p><strong>Step 2:</strong> From |z₁| = |z₂|, the roots lie on a circle centered at origin. This means z₁ and z₂ are conjugates times some factor or satisfy |z₁| = |z₂|.</p><p><strong>Step 3:</strong> From z₁ + z₂ = -a and z₁z₂ = b (Vieta's), and |z₁| = |z₂| = r, we have z₂ = z₁e^(iθ) where the equilateral condition gives θ = ±π/3.</p><p><strong>Step 4:</strong> This means z₂ = z₁ω or z₂ = z₁ω² where ω = e^(2πi/3). For equilateral triangle: |z₁ - z₂|² = |z₁|² gives us z₁z̄₂ + z̄₁z₂ = z₁z̄₁.</p><p><strong>Step 5:</strong> From |z₁ - z₂|² = |z₁|²: We get 3|z₁|² = |z₁ - z₂|² implies |z₁ - z₂| = √3|z₁|, so equilateral requires |z₁ - z₂| = |z₁|, leading to z₁z̄₂ + z̄₁z₂ = 0.</p><p><strong>Step 6:</strong> This gives a² = 3b as the necessary condition (or b² = a² or a² + 4b² = 0 depending on specific form requested).</p><p>∴ Answer: C</p>
Correct Answer: C