2021 MI P1 Q8

A curve \(C\) has parametric equations

\(x=\cot t+2\), \(y=\sec t\), \(-\frac{\pi }{2} < t < 0\)

1. Sketch the graph of \(C\), indicating the equations of any asymptotes.[2]
2. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{{{\sin }^{3}}t}{{{\cos }^{2}}t}\). Hence explain why \(C\) is increasing for \(-\frac{\pi }{2} < t < 0\).[3]
3. Find the equation of normal to \(C\) when \(t=-\frac{\pi }{4}\).[2]
4. The point \(P\) on \(C\) has coordinates \(\left( \cot p+2,\,\,\sec p \right)\). Given that the point \(R\) is the midpoint of \(P\) and the point with coordinates \(\left( -2,0 \right)\), find the cartesian equation of the curve traced by \(R\) as \(p\) varies.[3]