Complex Numbers
Modulus and Geometric Interpretation
Grade 11
Question:
<p>If \(|z - 2i| = \sqrt{2}\), where \(i = \sqrt{-1}\), then the maximum value of \(|3 + i(z - 1)|\) is</p>
<p>(a) \(2\)</p>
<p>(b) \(2\sqrt{2}\)</p>
<p>(c) \(2 + \sqrt{2}\)</p>
<p>(d) \(3 + 2\sqrt{2}\)</p>
Step-by-Step Solution
Key Concept: Use the geometric interpretation of modulus constraints. The locus $|z - 2i| = \sqrt{2}$ is a circle, and we maximize the distance from a point to this circle.
<p><strong>Given:</strong> $|z - 2i| = \sqrt{2}$</p><p><strong>Step 1:</strong> We have $|3 + i(z - 1)| = |i||z - 1 + 3i| = |z - 1 - 3i|$</p><p><strong>Step 2:</strong> Using the triangle inequality: $|z - 1 - 3i| \leq |z - 2i| + |-1 - i|$</p><p><strong>Step 3:</strong> From the constraint $|z - 2i| = \sqrt{2}$, we get $|z - 1 - 3i| \leq \sqrt{2} + |{-1 - i}|$</p><p><strong>Step 4:</strong> Since $|z - 2i| = \sqrt{2}$ defines a circle, the maximum value of $|z - 1 - 3i|$ occurs when $z$ is positioned optimally on this circle.</p><p><strong>Step 5:</strong> The maximum value is $2\sqrt{2}$.</p><p>∴ Answer is (b).</p>
Correct Answer: b