The distance of point `A(-2,3,1)` from the line `P Q` through `P(-3,5,2),` which makes equal angles with the axes is a. `2//sqrt(3)` b. `sqrt(14//3)` c. `16//sqrt(3)` d. `5//sqrt(3)`

Options

A. `2//sqrt3`
B. `sqrt(14//3)`
C. `16//sqrt3`
D. `5//sqrt3`

Answer

B

Explanation

Here, `alpha=beta=gamma`
`becausecos^(2)alpha+cos^(2)beta+cos^(2)gamma=1`
`becausecosalpha=(1)/(sqrt3)`
Direction cosines of PQ are `((1)/(sqrt3),(1)/(sqrt3),(1)/(sqrt3))`

PM=Projection of AP on PQ
`=|(-2+3)(1)/(sqrt3)+(3-5)(1)/(sqrt3)+(1-2)(1)/(sqrt3)|`
`=(2)/(sqrt3)`
and `AP=sqrt((-2+3)^(2)+(3-5)^(2)+(1-2)^(2))=sqrt6`
`AM=sqrt((AP)^(2)-(PM)^(2))=sqrt(6-(4)/(3))=sqrt((14)/(3))`