<p>If \(z\) lies on the circle \(|z - 2i| = 2\sqrt{2}\), then the value of \(\arg\left[\frac{z-2}{z+2}\right]\) is equal to</p>
Step-by-Step Solution
Key Concept: Parameterize z on the circle as z = 2i + 2√2·e^(iθ), then express (z-2)/(z+2) in terms of θ to find the argument, recognizing that the argument remains constant for all points on the circle.
<p><strong>Step 1:</strong> Parameterize the circle |z - 2i| = 2√2. Any point z on this circle can be written as:</p><p>z = 2i + 2√2·e^(iθ) = 2i + 2√2(cos θ + i sin θ)</p><p><strong>Step 2:</strong> Compute z - 2 and z + 2:</p><p>z - 2 = (2√2 cos θ - 2) + i(2√2 sin θ + 2)</p><p>z + 2 = (2√2 cos θ + 2) + i(2√2 sin θ + 2)</p><p><strong>Step 3:</strong> Find (z-2)/(z+2). Multiply numerator and denominator by the conjugate of denominator:</p><p>Let u = 2√2 cos θ + 2 and v = 2√2 sin θ + 2</p><p>Then: (z-2)/(z+2) = [(u-4) + iv][u - iv]/[u² + v²]</p><p><strong>Step 4:</strong> Notice the circle passes through specific points. When z = 2i (θ = π/2): z - 2 = -2 + 2i and z + 2 = 2 + 2i</p><p>(z-2)/(z+2) = (-2+2i)/(2+2i) = (-1+i)/(1+i) = [(-1+i)(1-i)]/[(1+i)(1-i)] = (-1+i+i+1)/2 = i</p><p><strong>Step 5:</strong> Verify this holds for another point, or use geometric insight: The locus relationship ensures arg[(z-2)/(z+2)] is constant = π/2</p><p>∴ Answer: B (which is π/2 or 90°)</p>
Correct Answer: B