2019 DHS P1 Q8

The equations of two planes \({{P}_{1}}\) and \({{P}_{2}}\) are \(x-2y+3z=4\) and \(3x+2y-z=4\) respectively.

1. The planes \({{P}_{1}}\) and \({{P}_{2}}\) intersect in a line \(L\). Find a vector equation of \(L\).[2]

The equation of a third plane \({{P}_{3}}\) is \(5x-ky+6z=1\), where \(k\) is a constant.

2. Given that the three planes have no point in common, find the value of \(k\).[2]

Use the value of \(k\) found in part (ii) for the rest of the question.

3. Given \(Q\) is a point of \(L\) meeting the \(x-y\) plane, find the shortest distance from \(Q\) to \({{P}_{3}}\).[3]
4. By considering the plane containing \(Q\) and parallel to \({{P}_{3}}\) or otherwise, determine whether the origin \(O\) and \(Q\) are on the same or opposite side of \({{P}_{3}}\).[2]