N2008 P1 Q11 [Modified]

The equations of three planes \({{p}_{1}}\), \({{p}_{2}}\), \({{p}_{3}}\) are

\(2x-5y+3z=3\)

\(3x+2y-5z=-5\)

\(5x+\lambda y+17z=\mu \)

respectively, where \(\lambda \) and \(\mu \) are constants. The planes \({{p}_{1}}\) and \({{p}_{2}}\) intersect in a line \(\ell \).

1. Find a vector equation of \(\ell \).
2. Given that the line \(\ell \) is contained in the plane \({{p}_{3}}\), find \(\lambda \) and \(\mu \).
3. Given instead that the line \(\ell \) does not intersect \({{p}_{3}}\), what can be said about the values of \(\lambda \) and \(\mu \)?
4. If the equation of \({{p}_{3}}\) is \(5x-22y+17z=5\), find the shortest distance between \(\ell \) and plane \({{p}_{3}}\).
5. Find the Cartesian equation of the plane which contains \(\ell \) and the point \((1,\,-1,\,3)\).