Question

If $|z-5i|$ = 5 and amp $z\in (0,\frac{\pi }{2}),$ then $\left[\cot \right(ampz)-\frac{10}{z}]$ is

• A. $-i$
• B. $1$
• C. $-1$
• D. $-\frac{1}{i}$

Answer 1

Answer: (D)

$-\frac{1}{i}$

$|z-5i|=5$ represents a circle, with center at $(0,5)$ and radius $5$. Evidently, $x$-axis is a tangent to the circle.
$P\left(z\right)$ is a point on the circle.
Let amp$\left(z\right)=\theta$
$\therefore$ From $\Delta OKP,$

$\sin \theta =\frac{OP}{10}\Rightarrow \left|z\right|=10\sin \theta$
$\therefore z=10\sin \theta (\cos \theta +i\sin \theta )$
$\frac{10}{z}=\frac{1}{\sin \theta }(\cos \theta -i\sin \theta )$
$\frac{10}{z}=(\cot \theta -i)$
$\Rightarrow \cot \theta -\frac{10}{z}=i=-\frac{1}{i}$