Question

If $|z-i|=1$ and arg $\left(z\right)=\theta$ where $\theta \in (0,\frac{\pi }{2})$, then
$cot\theta -\frac{2}{z}$

• A. 2i
• B. -i
• C. i
• D. 1+i

Answer 1

The correct option is C

i

$|z-i|=1$ represents a circle with centre at $(0,1)$ and radius 1. As $0<\theta <\frac{\pi }{2},\theta$ lies the semi-circle in the first quadrant.
As $\angle O=\frac{\pi }{2}$ and $\angle POA=\frac{\pi }{2}-\theta$, we get
$cos(\frac{\pi }{2}-\theta )=\frac{OP}{OA}\Rightarrow \sin \theta =\frac{|z|}{2}$
Also, since $\angle POA=\frac{\pi }{2}-\theta$
$\frac{2i}{z}=\frac{2i\overline{z}}{|z{|}^2}=\frac{2i|z|}{|z{|}^2}(\cos \theta -i\sin \theta )=\frac{2}{|z|}[cos(\frac{\pi }{2}-\theta )+isin(\frac{\pi }{2}-\theta )]=\frac{1}{sin\theta }[sin\theta +icos\theta ]=1+icot\theta \Rightarrow -\frac{2}{z}=i-cot\theta \Rightarrow cot\theta -\frac{2}{z}=i$