Question

If $|z-1|=1$ and $arg\left(z\right)=\theta$, where $z\ne 0$ and $\theta$is acute, then $1-\frac{2}{z}$ is equal to

• A. $\tan \theta$
• B. $i\tan \theta$
• C. $-\tan \frac{\theta }{2}$
• D. $\tan \frac{\theta }{2}$

Answer 1

The correct option is B $i\tan \theta$

$|z-1|=1$ represents a circle with centre at $1$ and radius equal to $1$.
We have $\angle OPA=\frac{\pi }{2}$
$\Rightarrow arg\left(\frac{2-z}{0-z}\right)=\frac{\pi }{2}\Rightarrow \frac{z-2}{z}=\frac{AP}{OP}i,$ Now in
$\Delta OAP,tan\theta =\frac{AP}{OP}$.Thus $\frac{z-2}{z}=itan\theta$