consider the mobius transformation $f(z) =\frac{1}{z} $ , $z \in \mathbb{C}$, $z \neq 0$.choose the correct options consider the  mobius transformation  $f(z) =\frac{1}{z} $  , $z \in \mathbb{C}$, $z \neq 0$. If C  denotes  a circle  with positive radius  passing through the  origin, then  $f$ map  $C \setminus \{0\}$ to

choose the correct options 

$1.$Circle

$2.$ a line

$3.$ a  line passing  through the  origin 

$4.$ a  line   not passing  through the  origin

My attempt :  i take  $z = e^{i\theta}$,$f(z) = 1/z  = 1/e^{i\theta}=e^{-i\theta}$

that $f(z)  =  \cos\theta  - i\sin\theta$

After that im  not  able  to proceed  further pliz help me

any hints/solution will be appreciated thanks u
What's the point of taking $z=e^{i\theta}$ that gives you a circle not passing through the origin? If you do take a circle passing through the origin, then what you will get is a line not passing through the origin. That is, the third option is the right one.
It is easy to show that $A(x^2+y^2)+Bx+Cy+D=0$ under $\frac1z$ goes to $D(u^2+v^2)+Bu-Cv+A=0$

So as the circle does not pass through the origin, $A$ and either $B$ or $C$ are non-zero and $D=0$

Hence, options 2 and 4 true.