8 A curve has equation
$$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 } ,$$
where \(k\) is a positive constant.
- Verify that
$$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$
are parametric equations for the curve.
- Hence or otherwise show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }\).
- Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-4_657_1071_938_577}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure} - Copy Fig. 8 and on the same axes sketch the curves for \(k = 1 , k = 3\) and \(k = 4\).
On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
- Explain why the path of the stream is modelled by the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$
- Solve this differential equation.
Given that the path of the stream passes through the point \(( 2,1 )\), show that its equation is \(y = \frac { x ^ { 4 } } { 16 }\).