Question

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

Answer 1

Let PQ be the perpendicular drawn from point P(2, 4, 3) to the line joining the points A(1, 2, 4) and B (3, 4, 5).

Let Q divides AB internally in the ratio λ:1

∴ Q ≡ `((3lambda + 1)/(lambda + 1), (4lambda + 2)/(lambda + 1), (5lambda + 4)/(lambda + 1))`   .......(i)

Direction ratios of PQ are

`(3lambda + 1)/(lambda + 1) - 2, (4lambda + 2)/(lambda + 1) - 4, (5lambda + 4)/(lambda + 1) - 3`

i.e., `(lambda - 1)/(lambda + 1), (-2)/(lambda + 1), (2lambda + 1)/(lambda + 1)`

Now, direction ratios of AB are, 3 – 1, 4 – 2, 5 – 4 i.e., 2, 2, 1.

Since PQ is perpendicular to AB,

`2((lambda - 1)/(lambda + 1)) + (2(-2))/(lambda + 1) + 1((2lambda + 1)/(lambda + 1))` = 0

∴ `(2lambda - 2 - 4 + 2lambda + 1)/(lambda + 1)` = 0

∴ 4λ − 5 = 0

∴ 4λ = 5

∴ λ = `5/4`

Putting λ = `5/4` in (i),

Coordinates of Q are,

`(3(5/4) + 1)/((5/4) + 1) = 19/9`

`(4(5/4) + 2)/((5/4) + 1) = 28/9`

`(5(5/4) + 4)/((5/4) + 1) = 41/9`

∴ Q ≡ `(19/9, 28/9, 41/9)`