Question

A complex variable $z=x+y\left(0.1\right)$ has its real part $x$ varying in the range $-\infty$ to $\infty$. Which one of the following is the locus (show in thick lines) of $\frac{1}{z}$ in the complex palne?

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let $z=x+iy$ and $w=u+iv$
$\Rightarrow z=\frac{1}{w}$
$\Rightarrow x+iy=\frac{1}{u+iv}=\frac{u-iv}{u^2+v^2}$
So, $x=\frac{u}{u^2+v^2}$ & $y=\frac{-v}{u^2+v^2}$
Now $y=0.1$
$\Rightarrow \frac{-v}{u^2+v^2}=\frac{1}{10}\Rightarrow u^2+v^2+10v=0$
$\Rightarrow u^2+(v+5{)}^2=25$.
Which is circle in w-plane having centre at $(0,-5)$ & radius $=5$