2025 RVHS Promo Q11

A curve \(C\) has parametric equations \[x=3t-\frac{3}{t},\ y=t+\frac{1}{t},\quad \text{where}\hspace{0.5em} t<0.\]

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{t^2-1}{3(t^2+1)}\).

   Hence, show the equation of the tangent to \(C\) at the point \(P\) with parameter \(p\) is

   \[3\left(p^2+1\right)y=\left(p^2-1\right)x+12p.\]

   [4]
2. Show that the cartesian equation of \(C\) may be written as \(9y^2-x^2=36\), stating the restriction on the values of \(y\).[3]
3. Sketch the graph of \(C\), giving the coordinates of any intercepts with the axes and the equations of any asymptotes.[3]

4. The region bounded by \(C\), the \(y\)-axis, the \(x\)-axis and the line \(x=4.5\) is rotated through \(2\pi\) radians about the \(y\)-axis. Find the volume of revolution of the solid formed, giving your answer correct to \(2\) decimal places.

   [3]