10 In this question you must show detailed reasoning.
Fig. 10 shows the curve given parametrically by the equations \(\mathrm { x } = \frac { 1 } { \mathrm { t } ^ { 2 } } , \mathrm { y } = \frac { 1 } { \mathrm { t } ^ { 3 } } - \frac { 1 } { \mathrm { t } }\), for \(t > 0\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7de77679-59c0-4431-a9cb-6ab11d2f9062-07_611_595_708_260}
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\caption{Fig. 10}
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- Show that \(\frac { d y } { d x } = \frac { 3 - t ^ { 2 } } { 2 t }\).
- Find the coordinates of the point on the curve at which the tangent to the curve is parallel to the line \(4 \mathrm { y } + \mathrm { x } = 1\).
- Find the cartesian equation of the curve. Give your answer in factorised form.