6. The transformation \(T\) maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation \(T\) is given by $$w = \frac { z } { i z + 1 } , \quad z \neq i$$ The transformation \(T\) maps the line \(l\) in the \(z\)-plane onto the line with equation \(v = - 1\) in the \(w\)-plane.

1. Find a cartesian equation of \(l\) in terms of \(x\) and \(y\). The transformation \(T\) maps the line with equation \(y = \frac { 1 } { 2 }\) in the \(z\)-plane onto the curve \(C\) in the \(w\)-plane.
2. 1. Show that \(C\) is a circle with centre the origin.
   2. Write down a cartesian equation of \(C\) in terms of \(u\) and \(v\).