8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-08_890_919_248_630}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of the curve \(C _ { 1 }\) with parametric equations
$$x = 2 \sin t , \quad y = 3 \sin 2 t \quad 0 \leq t < 2 \pi$$
- Show that the Cartesian equation of \(C _ { 1 }\) can be expressed in the form
$$y ^ { 2 } = k x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$
where \(k\) is a constant to be found.
The circle \(C _ { 2 }\) with centre \(O\) touches \(C _ { 1 }\) at four points as shown in Figure 5.
- Find the radius of this circle.