Question

Find the co-ordinates of the foot of the perpendicular drawn from the point A(–2, 3) to the line 3x – y – 1 = 0

Answer 1

Let M be the foot of perpendicular drawn from point A(– 2, 3) to the line

3x – y – 1 = 0 ....(i)

Slope of the line 3x – y – 1 = 0 is `(-3)/(-1)` = 3.

Since AM ⊥ to line (i),

slope of AM = `(-1)/3`

∴ Equation of AM is

y − 3 = `(-1)/3` (x + 2)

∴ 3(y − 3) = −1(x + 2)

∴ 3y − 9 = −x − 2

∴ x + 3y − 7 = 0 ...(ii)

The foot of perpendicular i.e., point M, is the point of intersection of equations (i) and (ii).

By (i) × 3 + (ii), we get

10x – 10 = 0

∴ x = 1

Substituting x = 1 in (ii), we get

1 + 3y – 7 = 0

∴ 3y = 6

∴ y = 2

The co-ordinates of the foot of the perpendicular M are (1, 2).