9 A curve has equation
$$y = \frac { x } { x - 1 }$$
- Find the equations of the asymptotes of this curve.
- Given that the line \(y = - 4 x + c\) intersects the curve, show that the \(x\)-coordinates of the points of intersection must satisfy the equation
$$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
- It is given that the line \(y = - 4 x + c\) is a tangent to the curve.
- Find the two possible values of \(c\).
(No credit will be given for methods involving differentiation.) - For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.