2016 NYJC Promo Q10

A point \(P\) with position vector \(2\mathbf{i}+\mathbf{j}-2\mathbf{k}\) lies on the line \(\ell \) with equation\(\frac{1-y}{3}=\frac{z+2}{4},x=2\) . The plane \({{\pi }_{1}}\) has equation \(x-2y-2z=8\).

1. Show that the coordinates of the point of intersection, \(Q\), of \(\ell \) and \({{\pi }_{1}}\) is \(\left( 2,7,-10 \right)\).[3]
2. Find the exact value of the sine of the acute angle between \(\ell \) and \({{\pi }_{1}}\).
   Hence find the exact length of projection of \(\overrightarrow{PQ}\) onto \({{\pi }_{1}}\).[4]

The plane \({{\pi }_{2}}\) is parallel to \({{\pi }_{1}}\). A point \(M\) has position vector \(2\mathbf{i}+\left( 4\alpha +4 \right)\mathbf{j}+\left( 3\alpha -6 \right)\mathbf{k}\) and is such that \(M\) is equidistant from \(\ell \) and .\({{\pi }_{2}}\)

3. Find the value(s) of \(\alpha\) if the distance from \(M\) to \(\ell \) is 10.[3]
   Hence deduce, when \(\alpha >0\) , the Cartesian equation of \({{\pi }_{2}}\).[3]