Q50 / 120 · VITEEE 2013 · Mathematics

A complex number $z$ is such that arg $\left(\frac{z-2}{z+2}\right)=\frac{z}{3}$. The points representing this complex number will lie on

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

Solution

$\begin{array}{l} \arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{3} \\ \Rightarrow \arg \left(\frac{x-2+i y}{x+2+i y}\right)=\frac{\pi}{3} \\ \Rightarrow \arg (x-2+i y)-\arg (x+2+i y)=\frac{\pi}{3} \\ \Rightarrow \tan ^{-1}\left(\frac{y}{x-2}\right)-\tan ^{-1}\left(\frac{y}{x+2}\right)=\frac{\pi}{3} \\ \Rightarrow \frac{4 y}{x^{2}+y^{2}-4}=\sqrt{3} \\ \Rightarrow \sqrt{3}\left(x^{2}+y^{2}\right)-4 y-4 \sqrt{3}=0 \end{array}$ which is an equation of a circle.

Hence, the correct answer is option: 'C'.

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