Question

What does $z+2$ represents, where $z$ is a complex number?

• A. All points $z+2$ lie on a circle with center $(-2,0)$ and radius $r$.
• B. All points $z+2$ lie on a circle with center $(2,0)$ and radius $r$.
• C. All points $z+2$ lie on a circle with center $(-2,0)$ and radius $2r$.
• D. All points $z+2$ lie on a circle with center $(2,0)$ and radius $2r$.

Answer 1

The correct option is B All points $z+2$ lie on a circle with center $(2,0)$ and radius $r$.

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