The curve \(C\) has equation
$$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$
where \(a\) is a constant not equal to 1,2 or 3 .
- Write down the equations of the asymptotes of \(C\).
- Show that \(C\) meets the asymptote parallel to the \(x\)-axis at the point where \(x = \frac { 2 a - 3 } { a - 2 }\).
- Show that the \(x\)-coordinates of any stationary points on \(C\) satisfy
$$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$
and hence find the set of values of \(a\) for which \(C\) has stationary points.
- Sketch the graph of \(C\) for
(a) \(a > 3\),
(b) \(2 < a < 3\).