Question

If $z$ is a complex number such that $Arg\left(\frac{z-1}{z-2}\right)=\frac{\pi }{4}$, then

• A. $z$ lies on a circle.
• B. $z$ lies on the major arc of a circle with centre $(2,2)$
• C. Length of path traced by $z$ by $\frac{3\pi }{2}units$
• D. none

Answer 1

The correct option is A $z$ lies on a circle.

We have,

$\left(\frac{z-1}{z-2}\right)=\frac{\pi }{4}\left(z-1\right)-\left(z-2\right)=\frac{\pi }{4}\left[\left(x-1\right)+iy-\left(x-2\right)+iy\right]=\frac{\pi }{4}$

Therefore,

${\tan }^{-1}\frac{y}{x-1}\left(\frac{xy-2y-xy+y}{x^2-3x+2+y^2}\right)=\frac{\pi }{4}xy-2y-xy+y=x^2-3x+2+y^2{x}^2+y^2-3x+y+2=0$

Hence, this equation is circle equation.