2017 AJC J1 MYE Q11

The attic of Winnie Pu’s house is in the form of a pyramid with a horizontal quadrilateral floor \(ABCD\).

Taking the apex as the origin \(O\), the position vectors of \(A\), \(B\), \(C\) and \(D\) are \(\left( \begin{matrix} 3 \\ 0 \\ -10 \\ \end{matrix} \right)\), \(\left( \begin{matrix} 6 \\ 8 \\ -10 \\ \end{matrix} \right)\), \(\left( \begin{matrix} -5 \\ 5 \\ -10 \\ \end{matrix} \right)\) and \(\left( \begin{matrix} -\sin \theta \\ -2\cos \theta \\ -10 \\ \end{matrix} \right)\) respectively, where \(0\le \theta \le \frac{\pi }{2}\).

The point \(E\) on the floor is such that \(OE\) is perpendicular to the floor, as shown in the diagram.

1. Find the maximum possible length of \(DE\) as \(\theta \) varies.[2]

A lamp with conical metal lampshade is hung from the apex such that its position vectors is \(-k\mathbf{k}\), where \(k\) is a positive constant. The semi-vertical angle of the lampshade is \(\frac{\pi }{3}\).

2. Winnie and his three friends each sits at one of the four corners \(A\), \(B\), \(C\) and \(D\) to read. For all four friends to read simultaneously, the light source needs to reach all four corners of the attic. Find the maximum value of \(k\) for the lamp to be hung.[3]

A honeybee flies into the attic and lands on a point \(H\) on the edge \(OC\).

3. Find the exact coordinates of \(H\) so that it is closest to the corner \(A\).[3]
4. Taking the lamp to be fixed at the maximum value of \(k\) found in part (ii), determine whether the honeybee is illuminated by the lamp.[4]