In R3,let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P1:x+2y−z+1=0 and .P2:2x−y+z−1=0 .Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1 . Which of the following points lie(s) on M?

Any line parallel to the line of intersection of planes P1≡x+2y−z+1=0and P2≡2x−y+z−1=0 is parallel to the vector                                                    i^j^k^12−12−11=i^−3k^−5k^∴Any point on the line L is in the form Aλ,−3λ,−5λ where λ∈RLet Mh,k,lbe the foot of the ⊥drawn from A onto the plane P1.Then          h−λ1=k+3λ2=l+5λ−1=−λ−6λ+5λ+11+4+1=−16⇒Mh,k,l=λ−16,−3λ−13,−5λ+16Clearly for λ=16,0, the points in options 1,2 lie on M.