2025 HCI P1 Q11

The points \(A\) and \(B\) have coordinates \((15, 3, 0)\) and \((5, 9, 5)\) respectively. Two lines \(l_1\) and \(l_2\), which are perpendicular to each other, have the following equations.

\(l_1 : \mathbf{r} = (15 - \lambda)\mathbf{i} + (3 + 2\lambda)\mathbf{j} + 4\lambda\mathbf{k},\)
\(l_2 : \mathbf{r} = (5 + 8\mu)\mathbf{i} + (9 - 2\mu)\mathbf{j} + (5 + m\mu)\mathbf{k},\)

where \(\lambda\) and \(\mu\) are parameters and \(m\) is a constant.

1. Find the value of \(m\).[2]
2. Given that \(l_1\) and \(l_2\) intersect, find the coordinates of the point of intersection \(E\).[2]
3. Find a cartesian equation of the plane \(\Pi\) which contains the points \(A\), \(B\) and \(E\).[3]
4. The point \(D\) has coordinates \((-1, -3, 2)\). Find the position vector of the point \(F\), the foot of perpendicular of \(D\) to \(\Pi\).[3]
5. Find the exact area of the circle that passes through \(A\), \(D\) and \(F\).[2]