Question

Find the coordinates of points on th line `(x - 1)/(1) =  (y - 2)/(-2) = (z - 3)/(2)` which are at the distance 3 unit from the base point A(l, 2, 3).

Answer 1

The cartesian equations of the line are `(x - 1)/(1) =  (y - 2)/(-2) = (z - 3)/(2) =  lambda`     ...(Say)

The coordinates of any point on this line are given by
x = λ + 1y = -2λ + 2z = 2λ + 3
Let M (λ + 1, -2λ + 2, 2λ + 3)              ...(1)
be the point on the in whose dstance from A(1, 2, 3) is 3 units.

∴ `sqrt(lambda + 1 - 1^2 + (-2lambda + 2 - 2)^2 + (2lambda + 3 - 3)^2`

∴ `sqrt(lambda^2 + 4lambda^2 + 4lambda^2)` = 3

∴ `sqrt(9lambda^2)` = 3

∴ `9lambda^2` = 9

∴ λ = ± 1

When λ = 1, M = (1 + 1, –2 + 2, 2 + 3)   ...[By (1)]

i.e. M = (2, 0, 5)

When λ = –1, M = (1 – 1, 2 + 2, –2 + 3)   ...[By (1)]

i.e. M = (0, 4, 1)

Hence, the coordinates of the required points are (2, 0, 5) and (0, 4, 1).