11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-32_858_743_118_603}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The curve \(C\) shown in Figure 3 has parametric equations
$$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leqslant t \leqslant 2 \pi$$
The curve \(C\) meets the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 3 .
- Write down the coordinates of \(A\) and \(B\).
- Find the values of \(t\) at which the curve passes through the origin.
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), and hence find the gradient of the curve when \(t = \frac { \pi } { 6 }\)
- Show that the cartesian equation for the curve \(C\) can be written in the form
$$y ^ { 2 } = a x ^ { 2 } \left( b - x ^ { 2 } \right)$$
where \(a\) and \(b\) are integers to be determined.