2026 HCI P1 Q8

The plane \(\Pi_1\) is defined by the equation \(\mathbf{r}\cdot((\beta-1)\mathbf{i}+\beta\mathbf{j}+2\beta\mathbf{k})=-3\), where \(\beta\) is a real constant. A point \(P\) has coordinates \((3,1,-2)\).

1. Verify that the point \(P\) lies on \(\Pi_1\).[1]
2. The plane \(\Pi_2\) contains \(P\) and is parallel to the vectors \(\begin{pmatrix}1\\1\\2\end{pmatrix}\) and \(\begin{pmatrix}1\\3\\4\end{pmatrix}\).
   Find, in terms of \(\beta\), the vector equation of the line \(l\) where \(\Pi_1\) and \(\Pi_2\) meet.[3]
3. The plane \(\Pi_3\) has cartesian equation \(x+2y=29-4z\).
   It is known that there is no common point of intersection between \(\Pi_1\), \(\Pi_2\) and \(\Pi_3\). Comment on the relationship between \(l\) and \(\Pi_3\). Hence find the value of \(\beta\).[2]
4. It is now given that \(\beta=3\).
   Given that a point \(A\) lies on \(\Pi_3\) and \(P\) is the foot of the perpendicular from \(A\) to \(\Pi_1\), find the coordinates of \(A\).[4]