The length of the perpendicular from P(1,6,3) to the line `x/1=(y-1)/(2)=(z-2)/(3) is `

Options

A. 3
B. `sqrt11`
C. `sqrt13`
D. 5

Answer

C

Explanation

Given line is `x/1=(y-1)/(2)=(z-2)/(3)`
`Rightarrow (x)/(1)=(y-1)/(2)=(z-2)/(3)=lambda`
`Rightarrow x=lambda, y=2lambda+1, z=3lambda+2`
`Rightarrow Q(lambda,2lambda+1, 3lambda+2)`
Therefore, direction ratios of PQ are
`(lambda-1,2lambda+1-6, 3lambda+2-3)`
`i.e, (lambda-1, 2lambda-5, 3lambda-1)`
`therefore` PQ is perpendicular to the given line. Therefore, `1(lambda-1)+2(2lambda-5)+3(3lambda-1)=0 Rightarrow lambda=1`
`therefore` The coordintes of Q are (1,3,5)
`therefore` Length of perpendicular
`PQ=sqrt((1-1)^(2)+(3-6)^(2)+(5-3)^(2))`
`=sqrt(9+4)=sqrt13`