Find the coordinates of the foot of theperpendicular drawn from the point `A(1,2,1)`to the line joining `B(1,4,6)a n dC(5,4,4)dot`

Find the coordinates of the foot of theperpendicular drawn from the po

1.	Find the coordinates of the foot of theperpendicular drawn from the point `A(1,2,1)`to the line joining `B(1,4,6)a n dC(5,4,4)dot`
Answer» The equation of a line joining the points P (1,4, 6) and (5,4,4) is `(x-1)/(5-1)=(y-4)/(4-4)=(z-6)/(4-6)` `(x-1)/(4)=(y-4)/(0)=(z-6)/(-2)" "...(1)` Let M be the foot of the perpendicular drawn from the point `(1,2,1).` Coordinates of M are given by `(x-1)/(4)=(y-4)/(0)=(z-6)/(-2)=lamda` `x=4lamda+1, y=0lamda+4, z=-2lamda+6` `therefore M-=(4lamda+1,4,-2lamda+6)" "...(2)` The direction ratios of AM are `4 lamda+1-1,4-2,-2lamda+6-1` i.e., `4lamda,2,-2lamda+5` Direction ratios of given line are `4,0,-2,` since AM is perpendicular to the given line `therefore4(4lamda)+0(2)+(-2)(-2lamda+5)=0` `therefore 16lamda+4lamda-10=0` `therefore lamda=1/2` Putting `lamda=1/2` in equation (2), we get `M-=(3,4,5)` Hence, the co-ordinates of the foot of the perpendicular are `(3,4,5)`