Complex Numbers
Modulus inequalities
Grade 11
Question:
<p>If \(z_1 = 5 + 12i\) and \(|z_2| = 4\), then</p>
<p>maximum \((|z_1 + iz_2|) = 17\)</p>
<p>minimum \((|z_1 + (1+i)z_2|) = 13 - 4\sqrt{2}\)</p>
<p>minimum \(\left|\dfrac{z_1}{z_2 + \dfrac{4}{z_2}}\right| = \dfrac{13}{4}\)</p>
<p>maximum \(\left|\dfrac{z_1}{z_2 + \dfrac{4}{z_2}}\right| = \dfrac{13}{3}\)</p>
Step-by-Step Solution
Key Concept: Use the triangle inequality |z₁ + z₂| ≤ |z₁| + |z₂| and |z₁ - z₂| ≥ ||z₁| - |z₂|| to establish bounds on expressions involving z₁ and z₂. Calculate |z₁| = √(25 + 144) = 13 first.
<p><strong>Step 1:</strong> Calculate |z₁|. Since z₁ = 5 + 12i, we have |z₁| = √(5² + 12²) = √(25 + 144) = √169 = 13.</p><p><strong>Step 2:</strong> Apply triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂| = 13 + 4 = 17, with equality when z₂ = kz₁ for k > 0.</p><p><strong>Step 3:</strong> Apply reverse triangle inequality: |z₁ + z₂| ≥ ||z₁| - |z₂|| = |13 - 4| = 9, with equality when z₂ = -kz₁ for k > 0.</p><p><strong>Step 4:</strong> Similarly, |z₁ - z₂| satisfies: 9 ≤ |z₁ - z₂| ≤ 17.</p><p>∴ Answer depends on specific options, but the bounds are: <strong>9 ≤ |z₁ ± z₂| ≤ 17</strong></p>
Correct Answer: ABCD