Question

The locus of $z$ in the argand diagram is the circle |z| $=$ $2,$ then show that the locus of $|z+1|$ is circle with centre $(1,0)$ and radius $2.$

Answer 1

Let
$z=2e^{i\theta }$
$=2cos\theta +i2sin\theta$
Hence
$z+1$
$=1+2cos\theta +i2sin\theta$
$=x+iy$
Then
$x-1=2cos\theta$
$y=2sin\theta$
Hence
$(x-1{)}^2+y^2=4$
Hence $z+1$ is the equation of a circle centered at $(1,0)$ with radius equal to 2.