2017 NYJC J1 MYE Q8

The diagram (not drawn to scale) shows three vertical flagpoles, \(OF\), \(AG\), \(BH\), with bases \(O\), \(A\), \(B\) respectively on an open and flat field, where \(OA=4\) metres and \(OB=8\) metres. The flagpoles have heights \(10\) metres, \(14\) metres and \(18\) metres respectively. The point \(O\) is taken as the origin, with unit vectors \(\mathbf{i}\) along \(OA\), \(\mathbf{j}\) along \(OB\) and \(\mathbf{k}\) along \(OF\).

The chairman of a neighbourhood committee wants to hold a songbird competition in the open field. He needs to erect a triangular flat shade sail using the flagpoles as supports and with fixing points at \(F\), \(G\) and \(H\).

1. Show that the cartesian equation of this shade sail is \(x+y-z=-10\).[3]

To strengthen the structure, he has to erect another vertical pole \(MN\) where the base \(M\) is at position vector \(2\mathbf{i}+3\mathbf{j}\).

2. Calculate the height of the pole if \(N\) is just touching the shade sail.[4]

In order to hang some bird cages, he needs to tie a taut rope from the midpoint of \(FH\) to a point \(P\) on \(GH.\)

3. Find the coordinates of \(P\) if the length of the rope is to be minimum,[4]
4. To help the residents find the location of the competition easily, the chairman ties a red helium balloon at \(H\) using another rope such that the balloon is vertically above \(H\). Assuming that there is no wind, find the angle between this rope and the shade sail.[2]