The curve \(C\) has equation
$$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$
where \(q\) is a positive constant.
- Obtain the equations of the asymptotes of \(C\).
- Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
- Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
- It is given that, for all values of the constant \(\lambda\), the line
$$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$
passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation
$$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$
has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).