<p>The roots of the equation \(t^3 + 3at^2 + 3bt + c = 0\) are \(z_1, z_2, z_3\) which represent the vertices of an equilateral triangle. Then</p>
Step-by-Step Solution
Key Concept: For an equilateral triangle centered at origin, roots satisfy z₁ + z₂ + z₃ = 0 (centroid condition). Use Vieta's formulas: sum of roots = -3a, which gives a = 0. The constraint |z₁ - z₂| = |z₂ - z₃| = |z₃ - z₁| combined with symmetry determines relationships between b and c.
<p><strong>Step 1:</strong> By Vieta's formulas for t³ + 3at² + 3bt + c = 0:</p><p>z₁ + z₂ + z₃ = -3a</p><p><strong>Step 2:</strong> For vertices of an equilateral triangle, the centroid (average of roots) must equal zero for the standard symmetric case. Thus z₁ + z₂ + z₃ = 0, which gives <strong>a = 0</strong>.</p><p><strong>Step 3:</strong> The equation reduces to t³ + 3bt + c = 0. For an equilateral triangle centered at origin with roots equally spaced at 120°, if |z| = r for each root, then roots are r·ω^k where ω = e^(2πi/3).</p><p><strong>Step 4:</strong> Using z₁z₂ + z₂z₃ + z₃z₁ = 3b: For roots r, rω, rω² we get 3b = r²(1 + ω + ω² + ω·ω² + ω²·r + r·ω) = r²(ω³ + ω + ω²) = 0 (since 1 + ω + ω² = 0).</p><p>Thus <strong>b = 0</strong>.</p><p><strong>Step 5:</strong> Using z₁z₂z₃ = -c: We get r³ω³ = r³ = -c, so <strong>c is real and c = -r³</strong>.</p><p>∴ Answer: <strong>a = 0, b = 0</strong> (or the constraint that relates b and c appropriately)</p>
Correct Answer: A