Use line and plane distances to determine a parallel plane

P has position vector $2\mathbf i+\mathbf j-2\mathbf k$ and lies on $\ell:(1-y)/3=(z+2)/4$, x=2. Plane π1 is $x-2y-2z=8$.

(i) Show the intersection Q of ℓ,π1 is (2,7,−10).3 marks

(ii) Find the exact sine of the acute angle between ℓ,π1. Hence find the exact length of projection of PQ onto π1.4 marks

Plane π2 is parallel to π1. M has position vector $2\mathbf i+(4\alpha+4)\mathbf j+(3\alpha-6)\mathbf k$ and is equidistant from ℓ and π2.

(iii) Find α if the distance from M to ℓ is 10. 3 marks Hence deduce, when α>0, the Cartesian equation of π2.3 marks