Question

arg $\left(\frac{z-2}{z-4}\right)=\frac{\pi }{3}$ represents

• A. equilateral triangle
• B. arc of circle
• C. arc of ellipse
• D. two rays making angle $\frac{\pi }{3}$ between them

Answer 1

The correct option is D two rays making angle $\frac{\pi }{3}$ between them

$arg\left(\frac{z-2}{z-4}\right)=arg(z-2)-arg(z-4)$

$arg\left(z\right)$ is angle which $z$ makes with positive $x-axis$,

$arg(z-2)$ is angle which $(z-2)$ makes with $x-axis$,

$arg(z-2)-arg(z-4)$ will refer to two imaginary numbers $(z-2)$ and $(z-4)$ having angle $\frac{\pi }{3}$ between them as $arg(z-2)-arg(z-4)=\frac{\pi }{3}$