Question

If $z=x+iy,r=\sqrt{x^2+y^2}=const.,$ what is the location of the points corresponding to
(i)$z+2$ (ii)$z-1+i$

Answer 1

We are given $z=x+iy$ and $\left|z\right|=r=const$.
(i) Let $Z=z+2$. Then $Z-2=z$
$\therefore |z-2|=\left|z\right|=r$
(1) shown that points corresponding to $Z=z+?$
lie on a circle of radius $r$ and center $(2,0)$.
(ii) Let $Z=z+1+i$. or $Z+1-i=z$
Then $|Z+1-i|=\left|z\right|=r$
(2) shown that the point representing $Z=z-1+?$ lie on a circle
of radius $r$ and having its centre the point $-1+i$ that is, at $(-1,1)$.