8 A curve has equation

$$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 } ,$$

where $k$ is a positive constant.\\
(i) Verify that

$$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$

are parametric equations for the curve.\\
(ii) Hence or otherwise show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }$.\\
(iii) Fig. 8 illustrates the curve for a particular value of $k$. Write down this value of $k$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-4_657_1071_938_577}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}

(iv) 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.\\
(v) Explain why the path of the stream is modelled by the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$

(vi) Solve this differential equation.

Given that the path of the stream passes through the point $( 2,1 )$, show that its equation is $y = \frac { x ^ { 4 } } { 16 }$.

\hfill \mbox{\textit{OCR MEI C4 2008 Q8 [18]}}