<p>If \(|z_1| = |z_2| = |z_3| = 1\) and \(z_1 + z_2 + z_3 = 0\), then the area of the triangle whose vertices are \(z_1, z_2, z_3\) is</p>
Step-by-Step Solution
Key Concept: Since |z₁| = |z₂| = |z₃| = 1, all three points lie on the unit circle. The constraint z₁ + z₂ + z₃ = 0 forces them to form an equilateral triangle centered at the origin.
<p><strong>Step 1:</strong> Since |z₁| = |z₂| = |z₃| = 1, all three complex numbers lie on the unit circle centered at the origin.</p><p><strong>Step 2:</strong> The constraint z₁ + z₂ + z₃ = 0 means the centroid of the triangle is at the origin. For three points on a circle with centroid at the center, they must form an equilateral triangle.</p><p><strong>Step 3:</strong> Let z₁ = e^(iθ), z₂ = e^(i(θ+2π/3)), z₃ = e^(i(θ+4π/3)). These vertices are separated by angles of 2π/3 (120°), confirming the equilateral triangle.</p><p><strong>Step 4:</strong> For an equilateral triangle inscribed in a unit circle with vertices at angular intervals of 2π/3, the side length is: s = |z₁ - z₂| = |e^(iθ) - e^(i(θ+2π/3))| = |1 - e^(i2π/3)| = √3</p><p><strong>Step 5:</strong> Area of equilateral triangle with side length s = √3: A = (s²√3)/4 = (3√3)/4</p><p>∴ Answer: <strong>3√3/4</strong></p>
Correct Answer: A