2026 RVHS P1 Q12

Underwater sonar buoys are deployed, in the ocean, to detect and monitor threats in the ocean and on the ocean surface. Two underwater sonar buoys, buoy A and buoy B, are located at points \(A\) and \(B\) respectively. The underwater sonar buoys are in the water body.

Points are defined relative to an origin \((0,0,0)\) on the ocean bed, with units in metres.

The ocean surface is modelled as the plane with equation \(3y+4z=240\). A submarine moves along a straight path with equation \(\frac x2=y-20=\frac{-z+13}2\). \(A\) and \(B\) have coordinates \((-10,31,5)\) and \((4,17,19)\) respectively.

1. Find a cartesian equation of the plane containing the path of the submarine and buoy B.[3]
2. Determine the acute angle between the path of the submarine and the ocean surface.[2]
3. An unidentified boat on the ocean surface is detected by buoys A and B. The boat is found to be equidistant from both buoys A and B. Describe geometrically the set of all possible positions of the location of the unidentified boat.[5]
4. At low tide, the ocean surface falls below the current ocean surface at a perpendicular distance of \(0.8\) metres. The ocean surface at low tide is also modelled as a plane. Find the cartesian equation of this plane.[2]