<p>If \(|w| = 1\), then \(\left|\dfrac{z}{z - \frac{i}{3}}\right| = 1\) implies that \(z\) lies on:</p>
Step-by-Step Solution
Key Concept: When |w| = 1 and |z/(z - i/3)| = 1, rewrite as |z| = |z - i/3| using the property |a/b| = |a|/|b|. This means z is equidistant from origin and point i/3, so z lies on the perpendicular bisector.
<p><strong>Step 1:</strong> Given |w| = 1 and |z/(z - i/3)| = 1</p><p><strong>Step 2:</strong> Using |a/b| = |a|/|b|, we get: |z|/|z - i/3| = 1</p><p><strong>Step 3:</strong> Therefore: |z| = |z - i/3|</p><p><strong>Step 4:</strong> This is the locus of points equidistant from the origin (0, 0) and the point i/3 (or 0, 1/3) on the imaginary axis.</p><p><strong>Step 5:</strong> The perpendicular bisector of the line segment joining 0 and i/3 is a horizontal line passing through the midpoint i/6 (or 0, 1/6).</p><p><strong>Step 6:</strong> The equation of this line is: Im(z) = 1/6, or equivalently y = 1/6</p><p>∴ z lies on the perpendicular bisector: a line parallel to the real axis at height 1/6 in the complex plane</p>
Correct Answer: C