Question

Find the locus of a complex number $z=x+iy$ satisfying the relation $|z+i|=|z+2|$. Illustrate the locus of z in the Argand plane.

Answer 1

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

assume $z=x+iy$

$x^2+(y+1{)}^2=(x+2{)}^2+y^2$

$x^2+y^2+2y+1=x^2+2x+y+y^2$

$2y+1=2x+1$

$x=y$

So, $z=z+ix$

$=x(1+i)$

So $z$ is always multiple pf $(1+i)$