2023 NYJC P1 Q10

At a theme park, the management decides to introduce a zipline adventure where a person can travel down from the top to the bottom of an inclined zipline on a pulley system, above a flat horizontal field.

Points are defined relative to an origin \(\left( 0,\mathrm{ }0,\mathrm{ }0 \right)\) on the flat horizontal field, where units are metres. The \(x\)-, \(y\)- and \(z\)-axes are in the directions of east, north and vertically upwards respectively. The zipline is above a flat horizontal field modelled by the equation \(z=0\). A take-off point is located at \(\left( 50,\mathrm{ }50,\mathrm{ }100 \right)\) on a vertical tower and an anchor point lies on another vertical tower as shown in the diagram below. The zipline is assumed to be taut at all times and the width of the zipline can be neglected.

1. The anchor point lies on a cable modelled by the equation \(\mathbf{r}=\left( \begin{matrix} 25 \\ 20 \\ 20 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 2 \\ 1 \\ 2 \\ \end{matrix} \right)\), \(\mu \in \mathbb{R}\). Given that the zipline is \(100\)m long with one end attached to the take-off point, find the coordinates of the anchor point, giving your answer correct to \(2\) decimal places.[4]

Due to safety reasons, the theme park management decides to relocate the anchor point and the new zipline follows an equation \(\mathbf{r}=\left( \begin{matrix} 50 \\ 50 \\ 100 \\ \end{matrix} \right)+\beta \left( \begin{matrix} 3 \\ 0 \\ 1 \\ \end{matrix} \right)\), \(\beta \in \mathbb{R}\).

2. Find the length of projection of the new zipline onto the field, given that the new zipline is also \(100\)m long.[3]
3. A flat safety net is to be constructed below the new zipline such that it is parallel to both the new zipline and the vector \(\left( \begin{matrix} 1 \\ 3 \\ 4 \\ \end{matrix} \right)\). At the same time, the flat safety net is to keep a distance of \(5\)m from the new zipline at all times. Find the cartesian equation modelled by the flat safety net.[3]
4. If the zipline can be modelled by the equation \(\mathbf{r}=\left( \begin{matrix} 50 \\ 50 \\ 100 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} a \\ 0 \\ c \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\), what can be said about the values of \(\lambda \), \(a\) and \(c\)?[2]