Find the length of the perpendicular from the point `(1,2,3)` to the line `(x-6)/3=(y-7)/2=(z-7)/(-2)dot`

Options

A. 5 units
B. 7 units
C. 4 units
D. none of these

Answer

B

Explanation

Let `L` be the foot of the perpendicular drawn from the point `P(1,2,3)` to the given line.
Let the coordinate of `L` be
`(3lamda+6,2lamda+7,-2lamda+7)`………….i
`:.` Direction ratos of PL are proportional to
`3lamda+6-1,2lamda+7-2,-2lamda+7-3`
i.e. `3lamda+5,2lamda+5,-2lamda+4`
Direction ratios of the given line are proportional to `3,2,-2`
Since PL is perpendicular to the given line.
`:.3(3lamda+5)+(2lamda+5)+(-2)(-2lamda+4)=0implieslamda=-1`

Putting `lamda=-1` in (i) we obtain that the coordinates of `L` as (3,5,9).
`:.PL=sqrt((3-1)^(2)+(5-2)^(2)+(9-3)^(2))=7` units