<p>Let \(z = x + iy\) be a complex number where \(x\) and \(y\) are integers. Then the area of the rectangle whose vertices are the roots of the equation \(z\bar{z}^3 + \bar{z}z^3 = 350\) is</p>
Step-by-Step Solution
Key Concept: Recognize that z·z̄³ + z̄·z³ = z̄z(z² + z̄²) = |z|²·2Re(z²), which reduces to finding |z| and relates the equation to real and imaginary parts. The four roots form a rectangle symmetric about both axes.
<p><strong>Step 1:</strong> Let z = x + iy where x, y ∈ ℤ. Then z̄ = x - iy and |z|² = x² + y².</p><p><strong>Step 2:</strong> Simplify z·z̄³ + z̄·z³ = z·z̄(z̄² + z²) = |z|²(z̄² + z²) = |z|²·2Re(z²).</p><p><strong>Step 3:</strong> Calculate z² = (x + iy)² = x² - y² + 2ixy, so Re(z²) = x² - y².</p><p><strong>Step 4:</strong> The equation becomes |z|²·2(x² - y²) = 350, giving (x² + y²)·2(x² - y²) = 350.</p><p><strong>Step 5:</strong> Simplify: 2(x⁴ - y⁴) = 350 ⟹ x⁴ - y⁴ = 175.</p><p><strong>Step 6:</strong> Factor: (x² - y²)(x² + y²) = 175 = 5 × 5 × 7. For integer solutions, try x² - y² = 5 and x² + y² = 35.</p><p><strong>Step 7:</strong> Solving: 2x² = 40 ⟹ x² = 20 (not a perfect square). Try x² - y² = 7 and x² + y² = 25.</p><p><strong>Step 8:</strong> Solving: 2x² = 32 ⟹ x² = 16 ⟹ x = ±4, and 2y² = 18 ⟹ y² = 9 ⟹ y = ±3.</p><p><strong>Step 9:</strong> The four roots are (±4, ±3), forming a rectangle with length 8 and width 6.</p><p><strong>Step 10:</strong> Area = 8 × 6 = 48.</p><p>∴ Answer: A</p>
Correct Answer: A