10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a6d0dba-d948-4124-9740-a88c17b0be65-32_556_716_237_607}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve \(C\) with parametric equations
$$x = \frac { 20 t } { 2 t + 1 } \quad y = t ( t - 4 ) , \quad t > 0$$
The curve cuts the \(x\)-axis at the point \(P\).
- Find the \(x\) coordinate of \(P\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - A ) ( 2 t + 1 ) ^ { 2 } } { B }\) where \(A\) and \(B\) are constants to be found.
- Make \(t\) the subject of the formula
$$x = \frac { 20 t } { 2 t + 1 }$$
- Hence find a cartesian equation of the curve \(C\). Write your answer in the form
$$y = \mathrm { f } ( x ) , \quad 0 < x < k$$
where \(\mathrm { f } ( x )\) is a single fraction and \(k\) is a constant to be found.