Question

$Ifz=x+iy,r=\sqrt{x^2+y^2}=$const., what is location of the points corresponding to $z+2$

• A. circle with center (-2,0) and radius r
• B. circle with center (2,0) and radius r
• C. circle with center (-2,0) and radius 2r
• D. circle with center (2,0) and radius 2r

Answer 1

The correct option is B circle with center (2,0) and radius r

Let $z=r{e}^{i\theta }$
$=r\cos \theta +i3\sin \theta$
Hence, $z+2$
$=r\cos \theta +2+i\left(r\sin \theta \right)$
$=x+iy$
Then,$x-2=r\cos \theta$
$y=r\sin \theta$
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 $r$.