Find the locus of a fixed-ratio point between a line and a plane

Line l is $x-1=(y+2)/(-1)=(z-1)/2$, and plane p1 has equation $\mathbf r=\alpha\mathbf i+(-\alpha+\beta)\mathbf j+(12+2\alpha-5\beta)\mathbf k$, α,β real.

(i) Show l is parallel to p1.1 mark

(ii) A,B vary independently on l,p1. C lies on segment AB with BC=2AC. Show C lies on a plane as A,B vary.3 marks

(iii) Calling this plane p2, show its equation is $\mathbf r\cdot(3,5,1)=0$.2 marks

(iv) Find the perpendicular distance between l,p2. Hence or otherwise find the distance between p1,p2.3 marks