Given that C is complex field . consider the mobius transformation $f(z) = \frac{1}{z} $, $z\in C$ , $z \neq 0$. if $D$ denotes a circle with positive radius passing through the origin, then f maps $D$ \ {$0$} to
1) a circle
2) a line
3) a line passing through the origin.
4) a line not passing through the origin.
my answer : i take $z = e^{iθ}$ now $ f(z) = e^{-iθ}$
now by theorem by Mobius Transformation theorem,,,it map Circle to circle...as D \ {0} is connected as continious image of connected set is connecteds...
so option 1) and option 3) is the only correct answer