8 A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\).
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- Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(t\).
- Hence show that the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at P is - 1 .
- Find the time at which the particle is travelling in the direction opposite to that at P .
- Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).