Question

If $|z-3|=3,$ then $\frac{z-6}{z}$ is equal to

• A. $i\tan \left(\arg z\right)$
• B. $i\cot \left(\arg z\right)$
• C. $\cot \left(\arg z\right)$
• D. $\tan \left[\arg (z-3)\right]$

Answer 1

Answer: (A)

$i\tan \left(\arg z\right)$

$|z-3|=3$ is a circle centred at the point $(3,0)$ and of radius $3$ . Hence, $Ais(6,0)$.
$AP=z-6,OP=z$
$OA$ subtends an angle of $\frac{\pi }{2}$ at $P$ . Now, $PA$ and $PO$ are of different lengths.
Hence, we take unit vectors and apply anticlockwise rotation.
$\frac{\overrightarrow{AP}}{AP}=\frac{\overrightarrow{OP}}{OP}{e}^{i\pi /2}$
$\frac{z-6}{z}=\frac{AP}{OP}.i=i\tan \theta =i\tan \left(\arg z\right)$