Question

If $arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$, then find the equation of the circle.

Answer 1

To find the equation of the circle :

Let $z=x+iy$

$$\begin{gathered} \Rightarrow arg\left(\frac{x+iy-1}{x+iy+1}\right)=\frac{\pi }{4} \\ \Rightarrow arg\left(\frac{(x-1)+iy}{(x+1)+iy}\times \frac{(x+1)-iy}{(x+1)-iy}\right)=\frac{\pi }{4} \\ \Rightarrow arg\left(\frac{(x^2-1)+y^2+i\left[\right(x+1)y-y(x-1\left)\right]}{{(x+1)}^2+y^2}\right)=\frac{\pi }{4} \\ \Rightarrow {\tan }^{-1}\left(\frac{xy+y-xy+y}{x^2-1+y^2}\right)=\frac{\pi }{4} \\ \Rightarrow {\tan }^{-1}\left(\frac{2xy}{x^2-1+y^2}\right)={\tan }^{-1}\left(1\right) \\ \Rightarrow \frac{2xy}{x^2-1+y^2}=1 \\ \Rightarrow x^2+y^2-2y-1=0 \\ \Rightarrow x^2+{(y-1)}^2={\left(\sqrt{2}\right)}^2 \end{gathered}$$

Hence, Centre $=(0,1)$ and radius $=\sqrt{2}$.