9 A curve has equation
$$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
- Find the equations of the three asymptotes of the curve.
- Show that if the line \(y = k\) intersects the curve then
$$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
- Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that
$$k ^ { 2 } - k \geqslant 0$$
- Hence show that the curve has only one stationary point and find its coordinates.
(No credit will be given for solutions based on differentiation.)
- Sketch the curve and its asymptotes.