Question

If $z=x+iy$ and arg$\left(\frac{z-2}{z+2}\right)=\frac{\pi }{6}$ then the locus of z is

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

Answer 1

Answer: (B)

circle

Put $z=x+iy$ in $arg\left(\frac{z-2}{z+2}\right)=\frac{\pi }{6}$

$arg\left(\frac{(x-2)+iy}{(x+2)+iy}\right)=\frac{\pi }{6}$

$\Rightarrow arg\left(\right(x-2)+iy)-arg\left(\right(x+2)+iy)=\frac{\pi }{6}$

$ta{n}^{-1}\frac{y}{x-2}-ta{n}^{-1}\frac{y}{x+2}=\frac{\pi }{6}$

$\Rightarrow ta{n}^{-1}\left(\frac{\frac{y}{x-2}-\frac{y}{x+2}}{1+\frac{y^2}{x^2-4}}\right)=\frac{\pi }{6}$

$\Rightarrow \frac{xy+2y-xy+2y}{x^2+y^2-4}=tan\frac{\pi }{6}=\frac{1}{\sqrt{3}}$

$=x^2+y^2-4\sqrt{3y}-4=0$, which is equation of a circle.