The curve \(C\) has equation
$$y = \frac { x ^ { 2 } + 2 \lambda x } { x ^ { 2 } - 2 x + \lambda }$$
where \(\lambda\) is a constant and \(\lambda \neq - 1\).
- Show that \(C\) has at most two stationary points.
- Show that if \(C\) has exactly two stationary points then \(\lambda > - \frac { 5 } { 4 }\).
- Find the set of values of \(\lambda\) such that \(C\) has two vertical asymptotes.
- Find the \(x\)-coordinates of the points of intersection of \(C\) with
(a) the \(x\)-axis,
(b) the horizontal asymptote. - Sketch \(C\) in each of the cases
(a) \(\lambda < - 2\),
(b) \(\lambda > 2\).