the distance of the point `(2,3,4)` from the line `(1-x)=y/2=1/3(1+z)`

Options

A. `(1)/(7)sqrt(35)`
B. `(4)/(7)sqrt(35)`
C. `(2)/(7)sqrt(35)`
D. `(3)/(7)sqrt(35)`

Answer

D

Explanation

Given,
`(x-1)/(-1)=(y-0)/(2)=(z+1)/(3)=r` [say] . . . (i)
Then coordinates of any point N on the line (i) are
`(-r+1,2r,3r-1)` . . . (ii)
Let N be the foot of the perpendicular to line (i).
`therefore` Direction ratios of PN are
`(-r+1-2,2r-3,3r-1-4)`
`=(-r-1,2r-3,3r-5)` . . . (iii)
`because` PN is perpendicular to line (i).
`therefore` Using the coordination,
`a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)=0`
`implies-1(-r-1)+2(2r-3)+3(3r-5)=0`
`impliesr+1+4r-6+9r-15=0`
`impliesr=(10)/(7)`
From Eq. (ii) we get
`N=(-(10)/(7)+1,(20)/(7),(30)/(7)-1)`
`=(-(3)/(7),(20)/(7),(23)/(7))`
Now, perpendicular distance
`PN=sqrt((-(3)/(7)-2)^(2)+((20)/(7)-3)^(2)+((23)/(7)-4)^(2))`
`=(1)/(7)sqrt(289+1+25)=(3)/(7)sqrt(35)`