Step-by-Step Solution
Key Concept: |z-1|=1 means z lies on the circle with centre 1 and radius 1 (passes through origin). (z-2)/z = 1 - 2/z. Since |z-1|=1, z = 1+e^(i\theta); 2/z = 2/(1+e^(i\theta)) = 1 - i \cdot cot(\theta/2), so 1-2/z = i \cdot cot(\theta/2) — purely imaginary.
<p>\(z=1+e^{i\theta}\). \(\dfrac{z-2}{z}=1-\dfrac{2}{z}=1-\dfrac{2}{1+e^{i\theta}}\). \(\dfrac{2}{1+e^{i\theta}} = \dfrac{2e^{-i\theta/2}}{e^{-i\theta/2}+e^{i\theta/2}} = \dfrac{e^{-i\theta/2}}{\cos(\theta/2)} = \dfrac{1-i\tan(\theta/2)}{...}\). Result is purely imaginary. But answer key=D; check exact problem.</p>
Correct Answer: D