<p>If \(\omega = \dfrac{2}{z - \dfrac{1}{3}i}\) and \(|\omega| = 1\), then \(z\) lies on</p>
Step-by-Step Solution
Key Concept: Since |ω| = 1, we have |2/(z - i/3)| = 1, which means |z - i/3| = 2. This is the equation of a circle with center at i/3 and radius 2.
<p><strong>Step 1:</strong> Given that ω = 2/(z - i/3) and |ω| = 1</p><p><strong>Step 2:</strong> Apply modulus to both sides: |ω| = |2/(z - i/3)| = 1</p><p><strong>Step 3:</strong> Use the property |a/b| = |a|/|b|: |2|/|z - i/3| = 1</p><p><strong>Step 4:</strong> Simplify: 2/|z - i/3| = 1, which gives |z - i/3| = 2</p><p><strong>Step 5:</strong> This represents a circle with center at the point (0, 1/3) in the complex plane and radius 2</p><p>∴ Answer: C (A circle with center i/3 and radius 2)</p>
Correct Answer: C