<p>\(Z \in \mathbb{C}\) satisfies the condition \(|Z| \geq 3\). Then find the least value of \(\left|Z + \dfrac{1}{Z}\right|\).</p>
Step-by-Step Solution
Key Concept: For |Z| ≥ 3, minimize |Z + 1/Z| by recognizing that the minimum occurs when Z is real and positive (Z = 3), then compute |3 + 1/3| = 10/3. However, the actual minimum on the boundary occurs when Z and 1/Z are optimally aligned; use calculus or parametrization to find the true minimum is 8/3 when |Z| = 3 and Z = 3e^(iθ) with specific θ.
<p><strong>Step 1:</strong> Let Z = re^(iθ) where r = |Z| ≥ 3.</p><p><strong>Step 2:</strong> Then Z + 1/Z = re^(iθ) + (1/r)e^(-iθ) = (r + 1/r)cos(θ) + i(r - 1/r)sin(θ)</p><p><strong>Step 3:</strong> |Z + 1/Z|² = (r + 1/r)²cos²(θ) + (r - 1/r)²sin²(θ)</p><p><strong>Step 4:</strong> Expanding: |Z + 1/Z|² = (r + 1/r)²cos²(θ) + (r - 1/r)²sin²(θ) = (r + 1/r)² - 4sin²(θ)</p><p><strong>Step 5:</strong> This is minimized when sin²(θ) = 1 (i.e., θ = π/2): |Z + 1/Z|² = (r + 1/r)² - 4</p><p><strong>Step 6:</strong> With r ≥ 3, the function f(r) = (r + 1/r)² - 4 is minimized at r = 3:</p><p>f(3) = (3 + 1/3)² - 4 = (10/3)² - 4 = 100/9 - 36/9 = 64/9</p><p><strong>Step 7:</strong> Therefore |Z + 1/Z|_min = √(64/9) = 8/3</p><p>∴ Answer: <strong>8/3</strong></p>
Correct Answer: 8/3