A line with positive diection cosines passes through the ont `P(2,-1,2) and ` makes equal angles with the coordinate axes. The line meets the plane `2x=y+z=0 ` at Q. The length of the line segment PQ equals (A) `1` (B) `sqrt(2)` (C) `sqrt(3)` (D) 2

Options

A. 1
B. `sqrt(2)`
C. `sqrt(3)`
D. 2

Answer

C

Explanation

Since, `l=m=n=(1)/(sqrt(3))`

`therefore ` Equation of line is,
`(x-2)/(1//sqrt(3))=(y+1)/(1//sqrt(3))=(z-2)/(1//sqrt(3))`
`rArr x-2=y+1=z-2=r` (say)
`therefore` Any point on the lne is,
`Q=(r+2, r-1,r+2)`
`because ` Q lies on the plane `2x+y+z=9`
`therefore 2(r+2)+(r-1)+(r+2)=9`
`rArr 4r+5=9`
`rArr r=1`
`therefore ` Coordinate Q(3, 0, 3)
`therefore PQ=sqrt(3-2)^(2)+(0+1)^(2)+(3-2)^(2))`
`=sqrt(3)`