If the image of the point `P(1,-2,3)` in the plane, `2x+3y-4z+22=0` measured parallel to the line, `x/1=y/4=z/5` is `Q` , then `P Q` is equal to : `sqrt(42)` (2) `6sqrt(5)` (3) `3sqrt(5)` (4) `3sqrt(42)`

Options

A. `sqrt(42)`
B. `6sqrt(5)`
C. `3sqrt(5)`
D. `2sqrt(42)`

Answer

D

Explanation

Let `R` be the point where the line passing through point `P(1,-2,3)` and parallel to the line `x/1=y/4=z/5` intersects the plane `2x+3y-4z+22=0`. Then `PQ=2PR`.

The equations of the line passing through `P(1,-2,3)` and parallel to `x/1=y/4=z/3` is `(x-1)/1=(y+2)/4=(z-3)/3`. Suppose it intersects the plane at `R`. The Coordinates of `R` are given by
`(x-1)/1=(y+2)4=(z-3)/5=lamda` or `(lamda+1,4lamda-2,5lamda+3)`
It lies on the plane `2x+3y-4z+22=0`
`:.2lamda+2+12lamda-6-20lamda-12+22=0implies-6lamda+6=0implieslamda=1`
So the coordinates of `R` are (2,2,8).
`:.PR=sqrt((2-1)^(2)+(2+2)^(2)+(8-3)^(2))=sqrt(42)`
Hence `PQ=2PR=2sqrt(42)`.