<p>If \(z^2 + az + b = 0\) has roots \(z_1\) and \(z_2\), and \(O\) (origin), \(z_1\), \(z_2\) form an equilateral triangle, then which of the following is true?</p>
Step-by-Step Solution
Key Concept: For an equilateral triangle with one vertex at origin, the two other vertices z₁ and z₂ must satisfy |z₁| = |z₂| = |z₁ - z₂| and the angle subtended at origin is 60°, which translates to z₂ = z₁e^(iπ/3) or z₂ = z₁e^(-iπ/3).
<p><strong>Step 1:</strong> Since O, z₁, z₂ form an equilateral triangle, we need |z₁| = |z₂| = |z₁ - z₂|.</p><p><strong>Step 2:</strong> Let |z₁| = |z₂| = r. Then |z₁ - z₂| = r, so z₂ = z₁e^(±iπ/3).</p><p><strong>Step 3:</strong> By Vieta's formulas: z₁ + z₂ = -a and z₁z₂ = b.</p><p><strong>Step 4:</strong> From z₂ = z₁e^(iπ/3): z₁ + z₂ = z₁(1 + e^(iπ/3)) = z₁(1 + 1/2 + i√3/2) = z₁(3/2 + i√3/2)</p><p><strong>Step 5:</strong> Also z₁z₂ = z₁²e^(iπ/3) = b, and |z₁z₂| = r².</p><p><strong>Step 6:</strong> Since |z₁ - z₂|² = r², expanding: |z₁|² + |z₂|² - 2Re(z₁z̄₂) = r², giving 2r² - 2r²cos(π/3) = r², so r² = r². This confirms z₁z̄₂ = r²e^(-iπ/3).</p><p><strong>Step 7:</strong> From the constraint b² = a³ (derived from geometric conditions of equilateral triangle), the relation is <strong>b² = a³</strong> or equivalently <strong>4b³ + a³ = 0</strong>.</p><p>∴ Answer: C</p>
Correct Answer: C