<p>If <\(\omega = z / [z - (1/3)i]\) and <\(|\omega| = 1\), then find the locus of <\(z\).</p>
Step-by-Step Solution
Key Concept: The locus of points satisfying |z| = |z - (1/3)i| represents all complex numbers equidistant from two fixed points: the origin and (1/3)i. This geometric condition defines a perpendicular bisector in the complex plane.
<p><strong>Step 1:</strong> Recognize that the equation |z| = |z - (1/3)i| means the distance from z to the origin equals the distance from z to the point (1/3)i.</p><p><strong>Step 2:</strong> The set of all points equidistant from two fixed points forms the perpendicular bisector of the line segment joining those points.</p><p><strong>Step 3:</strong> The midpoint of the segment joining 0 and (1/3)i is (1/6)i. The line segment is vertical (along the imaginary axis), so the perpendicular bisector is horizontal: Im(z) = 1/6, or the line y = 1/6 in the complex plane.</p><p><strong>Verification:</strong> Squaring both sides: |z|² = |z - (1/3)i|² gives x² + y² = x² + (y - 1/3)², which simplifies to y = 1/6.</p><p>∴ <strong>Answer:</strong> Perpendicular bisector of the line joining 0 + 0i and 0 + (1/3)i (the horizontal line Im(z) = 1/6)</p>
Correct Answer: Perpendicular bisector of the line joining 0 + 0i and 0 + (1/3)i