5 A curve has parametric equations \(x = \frac { t ^ { 2 } } { 1 + t ^ { 2 } } , y = t ^ { 3 } - \lambda t\), where \(\lambda\) is a constant.
- Use your calculator to obtain a sketch of the curve in each of the cases
$$\lambda = - 1 , \quad \lambda = 0 \quad \text { and } \quad \lambda = 1 .$$
Name any special features of these curves.
- By considering the value of \(x\) when \(t\) is large, write down the equation of the asymptote.
For the remainder of this question, assume that \(\lambda\) is positive.
- Find, in terms of \(\lambda\), the coordinates of the point where the curve intersects itself.
- Show that the two points on the curve where the tangent is parallel to the \(x\)-axis have coordinates
$$\left( \frac { \lambda } { 3 + \lambda } , \pm \sqrt { \frac { 4 \lambda ^ { 3 } } { 27 } } \right)$$
Fig. 5 shows a curve which intersects itself at the point ( 2,0 ) and has asymptote \(x = 8\). The stationary points A and B have \(y\)-coordinates 2 and - 2 .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-4_791_609_1482_769}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{figure} - For the curve sketched in Fig. 5, find parametric equations of the form \(x = \frac { a t ^ { 2 } } { 1 + t ^ { 2 } } , y = b \left( t ^ { 3 } - \lambda t \right)\), where \(a , \lambda\) and \(b\) are to be determined.