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If ω = (z)/(z - (i)/(3)) and |ω| = 1, then the point z must lie on

1. an ellipse
2. a circle
3. a straight line
4. a parabola

Correct answer: a straight line

Solution

With w=z/(z-i/3) and |w|=1, we get |z|=|z-i/3|, i.e. z is equidistant from 0 and i/3. That locus is the perpendicular bisector of the segment joining them, a straight line. Answer: option 2.

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