Question

The points representing the complex number z for which arg $\left(\frac{z-2}{z+2}\right)=\frac{\pi }{3}$ lie on

• A. a circle
• B. a straight line
• C. an ellipse
• D. a parabola

Answer 1

Answer: (A)

Let $z=x+iy$
$\therefore \frac{z-2}{z+2}=\frac{x+iy-2}{x+iy+2}$
$=\frac{(x-2)+iy}{(x+2)+iy}\times \frac{(x+2)-iy}{(x+2)-iy}$
$=\frac{(x-2)(x+2)+iy(x+2)-iy(x-2)-i^2{y}^2}{(x+2{)}^2-(iy{)}^2}$
$=\frac{x^2-4+ixy+2iy-ixy+2iy+y^2}{(x+2{)}^2+y^2}$
$=\frac{\left(x^2+y^2-4\right)+4iy}{(x+2{)}^2+y^2}=\frac{x^2+y^2-4}{\left(x^2+4+4x+y^2\right)}$
$+\frac{4iy}{x^2+4+4x+y^2}$
$\therefore \arg \left(\frac{z-2}{z+2}\right)={\tan }^{-1}\frac{4y}{x^2+4+4x+y^2}$
$\times \frac{x^2+4+4x+y^2}{x^2+y^2-4}=\frac{\pi }{3}$ (given)
$\therefore \tan \frac{\pi }{3}=\frac{4y}{x^2+y^2-4}$
$\sqrt{3}=\frac{4y}{x^2+y^2-4}$
$x^2+y^2-4-\frac{4}{\sqrt{3}}y=0$, which represents $a$ circle.