If the line` (x-2)/(3)=(y+1)/(2)=(z-1)/(-1)` intersects the plane `2x+3y-z+13=0` at a point P and the plane `3x+y+4z=16` at a point Q, then PQ is equal to

Options

A. 14
B. `sqrt(14)`
C. `2sqrt(7)`
D. `2sqrt(14)`

Answer

D

Explanation

Equation of given line is
`(x-2)/(3)=(y+1)/(2)=(z-1)/(-1)=r("let")" "...(i)`
Now, coordinates of a general point over given line is `R(3r+2,2r-1,-r+1)`
Let the coordinates of point P are `(3r_(1)+2,2r_(1)-1,r_(1)+1)` and Q are `(3r_(2)+2,2r_(2)-1,-r_(2)+1).`
Since, P is the point of intersection of line (i) and the plane `2x+3y-z+13=0,` so
`2(3r_(1)+2)+3(2r_(1)-1)-(-r_(1)+1)+13=0`
`implies" "6r_(1)+4+6r_(1)-3+r_(1)-1+13=0`
`implies" "13r_(1)+13=0impliesr_(1)=-1`
So, point `P(-1, -3, 2)`
And, similarly for point 'Q', we get
`3(3r_(2)+2)+(2r_(2)-1)+4(-r_(2)+1)=16`
`implies" "7r_(2)=7impliesr_(2)=1`
So, point is Q (5, 1, 0)
Now, `" "PQ=sqrt((5+1)^(2)+(1+3)^(2)+2^(2))`
`=sqrt(36+16+4)`
`=sqrt(56)=2sqrt(14)`