2022 EJC Promo Q12

A drone flies in a straight line from point \(A\) to point \(B\) with coordinates \(\left( 40,\,40,\,5 \right)\). At \(B\), it makes a sharp \({{90}^{\circ }}\) turn and thereafter flies in a straight line in the direction of \(2\mathbf{i}+\mathbf{j}+2\mathbf{k}\).

Eventually, it crashes into a cliff face which is a part of the plane \(8x+4y+z=595\), at point \(C\) (see diagram).

1. Find the coordinates of \(C\).[3]
2. Determine the acute angle \(\alpha \), that the path of the drone makes with the cliff face at \(C\).[2]
3. Given that the coordinates of \(A\) are \(\left( q,\,120,\,1 \right)\), find the value of \(q\).[3]
4. Max is trying to track the drone using his radar. From point \(D\) with coordinates \(\left( 36,\,36,\,0 \right)\), his radar emits constant signals in the directions of \(\cos \theta \,\mathbf{i}+\sin \theta \mathbf{j}+\mathbf{k}\) for all values of \(\theta \). Assuming that signals travel indefinitely along straight lines, show that none of the signals emitted succeed in hitting the drone as it flies from \(B\) to \(C\).[5]