<p>Find the greatest and the least value of \(|z_1 + z_2|\) if \(z_1 = 24 + 7i\) and \(|z_2| = 6\).</p>
Step-by-Step Solution
Key Concept: Use the triangle inequality: |z₁ + z₂| is maximized when z₂ points in the same direction as z₁ (adding magnitudes), and minimized when z₂ points opposite (subtracting magnitudes). This requires finding |z₁| first.
<p><strong>Step 1:</strong> Find |z₁|.</p><p>|z₁| = |24 + 7i| = √(24² + 7²) = √(576 + 49) = √625 = 25</p><p><strong>Step 2:</strong> Apply triangle inequality for |z₁ + z₂|.</p><p>By the triangle inequality: ||z₁| - |z₂|| ≤ |z₁ + z₂| ≤ |z₁| + |z₂|</p><p><strong>Step 3:</strong> Calculate the bounds.</p><p>Maximum value = |z₁| + |z₂| = 25 + 6 = 31 (when z₂ is in same direction as z₁)</p><p>Minimum value = ||z₁| - |z₂|| = |25 - 6| = 19 (when z₂ is in opposite direction as z₁)</p><p>∴ Greatest value = <strong>31</strong>, Least value = <strong>19</strong></p>
Correct Answer: Greatest value = 31, Least value = 19