Question

Find the points of the plane which satisfy the following equation:
$|z-2|=|z+2i|$

Answer 1

$|z-2|=|z+2i|$

${|z-2|}^2={|z+2i|}^2$

Let$z=x+iy$

${(x-2)}^2+y^2=x^2+{(y+2)}^2$

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

$⟹4-2x=4+4y$

$⟹4y=-2x$

$⟹y=-x/2$

Straight line passes through origin with slope$=-1/2$