- The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\) is given by
$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$
The straight line in the \(z\)-plane with equation \(y = 4 x\) is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane.
- Show that \(C\) has equation
$$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$
where \(k\) is a constant to be determined.
- Hence determine
- the coordinates of the centre of \(C\)
- the radius of \(C\)