<p>For any complex number \(z\), find the minimum value of \(|z| + |z - 2i|\).</p>
Step-by-Step Solution
Key Concept: The sum |z| + |z - 2i| represents the sum of distances from z to the origin and to the point 2i. By the triangle inequality, this sum is minimized when z lies on the line segment connecting these two points.
<p><strong>Step 1:</strong> Recognize that |z| is the distance from z to origin O(0,0), and |z - 2i| is the distance from z to point A(0,2).</p><p><strong>Step 2:</strong> The expression |z| + |z - 2i| represents the sum of distances from z to two fixed points O and A.</p><p><strong>Step 3:</strong> By the triangle inequality, for any point z: |z| + |z - 2i| ≥ |A - O| = |2i| = 2.</p><p><strong>Step 4:</strong> Equality holds when z lies on the line segment OA, i.e., when z = ti for t ∈ [0, 2]. At any point on this segment, |z| + |z - 2i| = |ti| + |ti - 2i| = t + |2 - t| = 2.</p><p><strong>Step 5:</strong> Therefore, the minimum value is achieved for all z on the segment from 0 to 2i, and equals |OA| = 2.</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2