<p>The roots of the cubic equation \((z + ab)^3 = a^3\), \(a \neq 0\), represent the vertices of a triangle of sides of length</p>
Step-by-Step Solution
Key Concept: The three cube roots of unity give three distinct solutions z = a(ω^k - b) where ω = e^(2πi/3). These three points form an equilateral triangle since they're related by 120° rotations around the center -ab.
<p><strong>Step 1:</strong> Solve (z + ab)³ = a³</p><p>Let w = z + ab, then w³ = a³</p><p>w = a·ω^k where ω = e^(2πi/3), k = 0, 1, 2</p><p><strong>Step 2:</strong> Find the three roots</p><p>z₁ = a(1) - ab = a(1 - b)</p><p>z₂ = a·e^(2πi/3) - ab = a(e^(2πi/3) - b)</p><p>z₃ = a·e^(4πi/3) - ab = a(e^(4πi/3) - b)</p><p><strong>Step 3:</strong> Calculate side lengths</p><p>|z₂ - z₁| = |a||e^(2πi/3) - 1| = |a|·|e^(πi/3)||e^(πi/3) - e^(-πi/3)| = |a|·2sin(π/3) = |a|√3</p><p>By symmetry (120° rotations): |z₃ - z₂| = |z₁ - z₃| = |a|√3</p><p><strong>Step 4:</strong> Identify the triangle</p><p>All three sides equal |a|√3, forming an equilateral triangle.</p><p>∴ Answer: B (sides of length <strong>|a|√3</strong> or <strong>a√3</strong>)</p>
Correct Answer: B