Question

The co-ordinates of the foot of perpendicular from the point A (1, 1, 1) on the line joining the points B (1, 4, 6) and C (5, 4, 4) are ______.

विकल्प

• (3, 4, 5)

• (4, 5, 3)

• (3, –4, 5)

• (–3, –4, 5)

Answer 1

The co-ordinates of the foot of perpendicular from the point A (1, 1, 1) on the line joining the points B (1, 4, 6) and C (5, 4, 4) are (3, 4, 5).

Explanation:

The equations of line BC are (using two point form)

`(x - 1)/(5 - 1) = (y - 4)/(4 - 4) = (z - 6)/(4 - 6)` i.e.,

`(x - 1)/2 = (y - 4)/0 = (z - 6)/(-1)`

Any point on this line is (2t + 1, 4, –t + 6). If this is the foot of perpendicular from A on the line BC, then d.n. of this perpendicular are

< 2t + 1 – 1, 4 – 1, –t + 6 – 1 > i.e. < 2t, 3, –t + 5 >

Using condition of perpendicularity, we have

(2t) (2) + 3 × 0 + (–t + 5) (–1) = 0

`\implies` 5t – 5 = 0

`\implies` t = 1

∴ Required foot of perpendicular is (2 + 1, 4, –1 + 6) = (3, 4, 5).