Question

Let $z$ be a complex number having the argument $\theta ,0<\theta <\pi /2$ and satisfying the equality $|z-3i|=3$. Then $\cot \theta -\frac{6}{z}$ is equal to

• A. $-i$
• B. $i$
• C. $2i$
• D. $1$

Answer 1

The correct option is B $i$

let, $z=r(\cos \theta +i\sin \theta ).$ Now,
$r=OA\sin \theta =6\sin \theta$
$⟹z=6\sin \theta (\cos \theta +i\sin \theta )$
or $\frac{6}{z}=\frac{1}{\sin \theta (\cos \theta +i\sin \theta )}$
$=\frac{\cos \theta -i\sin \theta }{\sin \theta }$
$=-i+\cot \theta$
or $cot\theta -\frac{6}{z}=i$
Ans: B