2022 JPJC J2 MYE P1 Q8

The plane \({{p}_{1}}\) has equation \(2x-y+3z=0\) and the line \(l\) has equation \(x=\frac{y-1}{a}=\frac{z-3}{b}\), where \(a\) and \(b\) are non-zero constants. The points \(A\) and \(B\) have position vectors \(9\mathbf{i}+2\mathbf{j}+4\mathbf{k}\) and \(\mathbf{j}+3\mathbf{k}\) respectively.

1. Given that line \(l\) is parallel to plane \({{p}_{1}}\), express \(a\) in terms of \(b\).[3]
2. Find the position vector of the foot of perpendicular from \(A\) to plane \({{p}_{1}}\) and hence find the shortest distance from \(A\) to plane \({{p}_{1}}\).[5]
3. Find the length of projection of \(\overrightarrow{OB}\) on the plane \({{p}_{1}}\).[3]

Another plane \({{p}_{2}}\) has equation \(cx-y+3z=d\).

4. Given that \(c=3\) and \(d=-1\), find a vector equation of the line of intersection between \({{p}_{1}}\) and \({{p}_{2}}\).[2]
5. Find the value of \(c\) and the range of values of \(d\) such that planes \({{p}_{1}}\) and \({{p}_{2}}\) are parallel and non-intersecting.[2]