A line `l` passing through the origin is perpendicular to the lines `l_1: (3+t)hati+(-1+2t)hatj+(4+2t)hatk , oo < t < oo , l_2: (3+s)hati+(3+2s)hatj+(2+s)hatk , oo < t < oo` then the coordinates of the point on `l_2` at a distance of `sqrt17` from the point of intersection of `l&l_1` is/are:

Options

A. `((7)/(3), (7)/(3), (5)/(3))`
B. `(-1, -1, 0)`
C. `(1, 1, 1)`
D. `((7)/(9), (7)/(9), (8)/(9))`

Answer

b, d

Explanation

The common perpendicular is along
`" "|{:(hati,,hatj,,hatk),(1,,2,,2),(2,,2,,1):}|=-2hati+3hatj-2hatk`
Let `" "M-=(2lamda, -3lamda, 2lamda)`
So, `" "(2lamda-3)/(1)= (-3lamda+1)/(2)= (2lamda-4)/(2)`
`rArr" "lamda=1`
So, `" "M-=(2, -3, 2)`
Let the required point be P.
Given the `PM=sqrt(17)`
`rArr" "(3+2s-2)^(2)+ (3+2s+3)^(2)+ (2+s-2)^(2)=17`
or `" "9s^(2)+ 28s+20=0 or s=-2, - (10)/(9)`.
So, `" "P-= (-1, -1, 0) or ((7)/(9), (7)/(9), (8)/(9))`