Question

Find the co-ordinates of the foot of the perpendicular drawn from the point (2, 3, −8) to the line `(4 - x)/(2) = y/6 = (1 - z)/(3)`.

Also, find the perpendicular distance of the given point from the line. 

Answer 1

Given line is `(4 - x)/(2) = y/6 = (1 - z)/(3)`

or, `(x - 4)/(-2) = y/6 = (z - 1)/(-3) = lambda`

Now, the co-ordinates of any points Q on the line are (−2λ + 4, 6λ, −3λ + 1).

Also, the given point is p(2, 3, −8)

The direction ratios of PQ are −2λ + 4 − 2, 6λ −3, − 3λ + 1 + 8 i.e., −2λ + 2, 6λ − 3, −3λ + 9

Also, the direction cosines of the given line are −2, 6, −3.

If PQ ⊥ line, then

−2(−2λ + 2) + 6(6λ − 3) − 3 (−3λ + 9) = 0

4λ − 4 + 36λ − 18 + 9λ − 27 = 0

49λ − 49 = 0

λ = 1

Now, the foot of the perpendicular is [−2(1) + 4, 6(1), −3(1) + 1] i.e., (2, 6, −2)

Hence, the distance PQ is 

= `sqrt((2 - 2)^2 + (3 - 6)^2 + (-8 + 2)^2)`

= `sqrt(0 + 9 + 36)`

= `sqrt45`

= `3sqrt5` units