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

Explanation

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)`