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. ` sqrt ( 11 ) `
C. ` sqrt ( 13 ) `
D. ` 5 `

Answer

C

Explanation

Given ` (x ) / ( 1 ) = ( y - 1 ) / ( 2 ) = ( z - 2 ) / ( 3 ) = lamda `
Any point on the line is ` P ( lamda , 2 lamda + 1 , 3 lamda + 2 ) `
Therefore, direction ratios of PQ are ` ( lamda - 1 , 2 lamda - 5, 3lamda - 1 ) `
` because PQ ` is perpendicular to the given line
Therefore, ` 1 ( lamda - 1 ) + 2 ( 2lamda - 5 ) + 3 ( 3lamda - 1 ) = 0 `
` " "[ because a - 1 a _ 2 + b _ 1 b _ 2 + c _ 1 c _ 2 = 0 ] `
` rArr lamda = 1 `
` therefore ` The coordinates of P are ` ( 1, 3, 5 )`
` therefore ` Length of perpendicular , QP
` = sqrt (( 1 - 1) ^ 2 + (3 - 6 ) ^ 2 + ( 5 - 3) ^ 2) = sqrt ( 13 ) `