Question

If z lies on $|z-2i|=2\sqrt{2}$ then arg $\left(\frac{z-2}{z+2}\right)$ is equal to

• A. $\frac{\pi }{8}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C $\frac{\pi }{4}$

$|z-2i|=2\sqrt{2}$ represents a circle with centre C (0, 2) and radius $2\sqrt{2}$
A (2, 0), B (- 2, 0) are points on the circle
Slope of CA . Slope of CB = - 1
$\therefore \angle ACB={90}^0$
then $\angle APB=\frac{1}{2}.\frac{\pi }{2}=\frac{\pi }{4},$ where P is any point on the circumference
$\Rightarrow arg\left(\frac{z-2}{z+2}\right)=\frac{\pi }{4}$