The shortest distance between the straight lines
`(x-6)/(1)=(2-y)/(2)=(z-2)/(2)" and "(x+4)/(3)=(y)/(-2)=(1-z)/(2)` is

Options

A. 9
B. `(25)/(3)`
C. `(16)/(3)`
D. 4

Answer

B

Explanation

Given lines can be rewritten as
`(x-6)/(1)=(y-2)/(-2)=(z-2)/(2)`
and `(x+4)/(3)=(y)/(-2)=(z-1)/(-2)`
Here, `x_(1)=6,y_(1)=2,z_(1)=2`
`x_(2)=-4,y_(2)=0,z_(2)=1`
`a_(1)=1,b_(1)=-2,c_(1)=2`
and `a_(2)=3,b_(2)=-2,c_(2)=-2`
Now, `|{:(x_(2)-x_(1),,y_(2)-y_(1),,z_(2)-z_(1)),(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)):}|`
`=|{:(-10,-2,-1),(" "1, -2, " "2),(" "3, -2, -2):}|`
`=-10(4+4)+2(-2-6)-1(-2+6)=-100`
and `sqrt((b_(1)c_(2)-c_(1)b_(2))^(2)+(c_(1)a_(2)-a_(1)c_(2))^(2)+sqrt((a_(1)b_(2)-a_(2)b_(1))^(2)))`
`=sqrt((4+4)^(2)+(6+2)^(2)+(-2+6)^(2))`
`=sqrt(64+64+16)=12`
`:.` Required shortest distance
`=(-100)/(12)=(25)/(3)" units "["neglect (-ve) sign"]`