9758/2022/P1/Q11

11) A gas company has plans to install a pipeline from a gas field to a storage facility. One part of the route for the pipeline has to pass under a river. This part of the pipeline is in a straight line between two points, $P$ and $Q$.

Points are defined relative to an origin $(0,0,0)$ at the gas field. The $x-, y$ – and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. $P$ has coordinates $(1136,92, p)$ and $Q$ has coordinates $(200,20,-15)$.
(a) The length of the pipeline $P Q$ is $939 \mathrm{~m}$. Given that the level of $P$ is below that of $Q$, find the value of $p$.



A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are $(400,600,-20),(500,200,-70)$ and $(600,-340,-50)$.
(b) Find the cartesian equation of this plane.



(c) Hence find the coordinates of the point where the pipeline meets the rock.



(d) Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.