Question

If the locus of the complex number $z$ given by $arg\left(z + i\right)-arg\left(z - i\right)=\frac{2 \pi }{3}$ is an arc of a circle, then the length of the arc is

• A. $\frac{4 \pi }{3}$
• B. $\frac{4 \pi }{3 \sqrt{3}}$
• C. $\frac{2 \sqrt{3} \pi }{3}$
• D. $\frac{2 \pi }{3 \sqrt{3}}$

Answer 1

Answer: (B)

Given, $arg\left(\frac{z + i}{z - i}\right)=\frac{2 \pi }{3}$
Let, $z=x+iy$
$\Rightarrow \frac{x + i \left(y + 1\right)}{x + i \left(y - 1\right)}=\frac{x^{2} + \left(y^{2} - 1\right) + 2 i x}{x^{2} + \left(y - 1\right)^{2}}$
$\Rightarrow tan^{- 1}\frac{2 x}{x^{2} + y^{2} - 1}=\frac{2 \pi }{3}$
$\Rightarrow x^{2}+y^{2}+\frac{2}{\sqrt{3}}x-1=0$
Hence, the given locus is a circle with centre $\left(- \frac{1}{\sqrt{3}} , 0\right)$ and radius $\frac{2}{\sqrt{3}}$ units

$\Rightarrow $ Length of the arc of the circle is $\frac{2 \pi }{3}\times \left(\frac{2}{\sqrt{3}}\right)=\frac{4 \pi }{3 \sqrt{3}}$ units