Question

The locus of the points z which satisfy the condition arg $\left( \frac{z - 1}{z + 1} \right) = \frac{\pi}{3}$ is

• A. A straight line
• B. A circle
• C. A parabola
• D. None of these

Answer 1

Answer: (C)

A parabola

Answer 2

Sol. We have $\frac{z - 1}{z + 1} = \frac{x + iy - 1}{x + iy + 1} = \frac{(x^{2} + y^{2} - 1) + 2iy}{(x + 1)^{2} + y^{2}}$

⇒ arg

$\frac{z - 1}{z + 1} = \tan^{- 1}\frac{2y}{x^{2} + y^{2} - 1}$

Hence $\tan^{- 1}\frac{2y}{x^{2} + y^{2} - 1} = \frac{\pi}{3}$

⇒ $\frac{2y}{x^{2} + y^{2} - 1} = \tan\frac{\pi}{3} = \sqrt{3}$

⇒ $x^{2} + y^{2} - 1 = \frac{2}{\sqrt{3}}y \Rightarrow x^{2} + y^{2} - \frac{2}{\sqrt{3}}y - 1 = 0$,

Which is obviously a circle