2025 CJC Promo Q11

The plane \(\Pi_1\) contains the points \(A(1,4,2)\), \(B(1,0,5)\) and \(C(0,8,-1)\).

1. Find a cartesian equation of \(\Pi_1\).[3]
2. Find the shortest distance between the point \(P(2,1,2)\) and the plane \(\Pi_1\).[2]

A second plane \(\Pi_2\) contains the point \(D(2,2,3)\) and is perpendicular to the vector \(\mathbf{i}+2\mathbf{j}+2\mathbf{k}\). The point \((p,0,q)\) lies in both planes.

4. Find \(p\) and \(q\).[3]
5. Hence find an equation of the line of intersection of the two planes in the form \(\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}\), where \(\lambda\) is a real constant.[2]
6. Find the acute angle between the line \(AC\) and the plane \(\Pi_2\).[2]