2025 TJC P1 Q11

At a campsite, an \(A\)-frame tent is pitched on level ground. Its roof is formed by two slanted polyester sheets lying on two intersecting planes,

\[\pi_1: x + y + az = 6\]

\[\pi_2: 2x - y + z = 4\]

where \(a\) is a positive real constant. The intersection of \(\pi_1\) and \(\pi_2\) forms a central ridge in the form of a line \(AB\). This ridge is extended with a rope, which is secured to the ground at the point \(C\) for stability. The ground is assumed to be the horizontal \(xy\)-plane.

1. Find a vector parallel to the central ridge, giving your answer in terms of \(a\).[2]
2. If \(C\) lies on \(AB\) produced, show that the coordinates of \(C\) is \(\left(\frac{10}{3}, \frac{8}{3}, 0\right)\).[3]
   Hence, or otherwise, write down a vector equation of line \(AB\) in term of \(a\).[1]
3. If the angle between the two slanted polyester sheets is \(60{}^\circ\), find the value of \(a\).[3]

It is now given that \(a = 1\).

4. A string is tied between two hooks inside the tent to hang decorations. One hook is fixed at the point \(D(2, 3, 1)\) which lies on \(\pi_1\). The other hook is located at the point \(E\) which lies on \(\pi_2\) such that the string \(DE\) is perpendicular to \(\pi_2\). Find the coordinates of \(E\).[3]