Question

The point P is the intersection of the straight line joining the points `Q(2, 3, 5)` and `R (1, -1, 4)` with the plane `5x- 4y-z=1`. If S is the foot of the perpendicular drawn from the point `T(2, 1, 4)` to QR, then the length of the line segment PS is:

Answer 1

`L:(x-1)/1=(y+1)/4=(z-4)/1=lambda`
`(lambda+1,4lambda-1,lambda+4)`
`5x-4y-z=1`
`5(lambda+1)-4(4lambda-1)-(lambda+4)=1`
`-12lambda+5=1`
`lambda=1/3`
`vec(TS)(lambda-1,4lambda-2,lambda)`
`(1,4,1)`
`lambda-1+16lambda-8+lambda=0`
`lambda=1/2`
`P(4/3,1,13/3),S(3/2,1,9/2)`
`PS=sqrt((3/2-4/3)^2+(9/2-13/3)^2)`
`PS=sqrt(1/6^2+1/6^2)`
`PS=sqrt2/6=1/(3sqrt2)`.