<p>If <i>|z₁| = 1, |z₂| = 2, |z₃| = 3</i> and <i>|9z₁z₂ + 4z₁z₃ + z₂z₃| = 36</i>, then <i>|z₁ + z₂ + z₃|</i> is equal to:</p>
Step-by-Step Solution
Key Concept: Use the constraint that |9z₁z₂ + 4z₁z₃ + z₂z₃| = 36 equals the maximum possible value, which occurs when all terms align in the same direction. This suggests z₁, z₂, z₃ are collinear (same argument), allowing us to treat them as positive real multiples.
<p><strong>Step 1:</strong> Assume z₁, z₂, z₃ are collinear with the same argument (say all positive reals or same direction in complex plane). Then z₁ = e^(iθ), z₂ = 2e^(iθ), z₃ = 3e^(iθ) for some angle θ.</p><p><strong>Step 2:</strong> Substitute into |9z₁z₂ + 4z₁z₃ + z₂z₃|:<br/>9z₁z₂ + 4z₁z₃ + z₂z₃ = 9(e^(iθ))(2e^(iθ)) + 4(e^(iθ))(3e^(iθ)) + (2e^(iθ))(3e^(iθ))<br/>= 18e^(2iθ) + 12e^(2iθ) + 6e^(2iθ) = 36e^(2iθ)</p><p><strong>Step 3:</strong> Therefore |9z₁z₂ + 4z₁z₃ + z₂z₃| = |36e^(2iθ)| = 36 ✓ (This confirms our assumption is correct)</p><p><strong>Step 4:</strong> Calculate |z₁ + z₂ + z₃|:<br/>z₁ + z₂ + z₃ = e^(iθ) + 2e^(iθ) + 3e^(iθ) = 6e^(iθ)</p><p><strong>Step 5:</strong> Therefore |z₁ + z₂ + z₃| = |6e^(iθ)| = 6</p><p><strong>∴ Answer:</strong> 6</p>
Correct Answer: 6