Question

If Amp $( \frac {z-1} {z+1})= \frac {\pi} {3}$, then $z$ represents a point on

• A. a pair of lines
• B. a straight line
• C. a circle
• D. none of these

Answer 1

Answer: (C)

a circle

Answer 2

$Amp \left(\frac{z-1}{z+1}\right)=\frac{\pi}{3}$ $\Rightarrow Amp\left(\frac{x-1+iy}{x+1+iy}\right)=\frac{\pi}{3}$ $\Rightarrow tan^{-1} \frac{y}{x-1}-tan^{-1} \frac{y}{x+1}=\frac{\pi}{3}$ $\Rightarrow tan^{-1} \frac{\frac{y}{x-1}-\frac{y}{x+1}}{1+\frac{y^{2}}{x^{2}-1}}=\frac{\pi}{3}$ $\Rightarrow \frac{y\left[x+1-x+1\right]}{x^{2}+y^{2}-1}=tan \frac{\pi}{3}=\sqrt{3}$ $\Rightarrow \frac{2y}{x^{2}+y^{2}-1}=\sqrt{3}$ $\Rightarrow x^{2}+y^{2}-1=\frac{2y}{\sqrt{3}}$ $\Rightarrow x^{2}+y^{2}-\frac{2y}{\sqrt{3}}-1=0$ which is a circle