Question

locate the point representing the complex numbers $z$ on the Argand diagram for which
$|z+i|=|z-2|$

Answer 1

$|z+i|=|z-2|$ or $ \left| z+i
\right| ^{ 2 }=\left| z-2 \right| ^{ 2 }$
or ${|x+iy+i|}^2={|x+iy-2|}^2$
or $x^2+(y+1{)}^2=(x-2{)}^2+y^2$
or $4x+2y-3=0$
Hence point $z$ satisfying the given equation lie on the
straight line $\left(1\right)$.
Alternative. We know that $\left| z+i \right| =\left| z-(-i)
\right| $isthedistancefromthepoint$z$ to the point
representing the number $-i$ and $|z+2|$ is the
distance from the point $z$ to the point representing the
number $2$. It is required to find the point for which these
distances are equal. The solution will thus be a locus of a point
equidistant from tow fixed point representing the number $-i$
and $2$ i.e from the point $(0,-1)$ and $(2,0)$.
From the geometry we know that this locus is a straight line
perpendicular to a line segment connecting the points $(0,-1)$
and $(2,0)$ and passing through its mid-point. Hence all point
$z$ satisfying $|z+i|=|z-2|$ lie on this line.