Question

The equation of the locus of $z$ such that $\left|\frac{(z-i)}{(z+i)}\right|=2$, where $z=x+iy$is a complex number is

Answer 1

Compute the required equation.

Given: $\left|\frac{(z-i)}{(z+i)}\right|=2$

$\Rightarrow$ $\left|(z-i)\right|=2\left|(z+i)\right|$

Take $z=x+iy$

$\Rightarrow$ $\left|x+i(y-1)\right|=2\left|x+i(y+1)\right|$

$\Rightarrow$ $x^2+{(y-1)}^2=4(x^2+{(y+1)}^2)$

$\Rightarrow$ $x^2+y^2-2y+1=4x^2+4y^2+8y+4$

$\Rightarrow$$3x^2+3y^2+10y+3=0$

Hence the equation of the locus is $3x^2+3y^2+10y+3=0$.