Answer the following question:

Find the co-ordinates of the foot of the perpendicular drawn from the point P(−1, 3) the line 3x − 4y − 16 = 0

#### Solution

Let M be the foot of perpendicular drawn from P(–1, 3) to the line

3x – 4y – 16 = 0 ...(i)

Slope of the line 3x – 4y – 16 = 0 is

`(-3)/(-4) = 3/4`

Since PM ⊥ to line (i),

slope of PM = `(-4)/3`

∴ Equation of PM is

y – 3 = `(-4)/3(x + 1)`

∴ 3(y – 3) = –4(x + 1)

∴ 3y – 9 = – 4x – 4

∴ 4x + 3y – 5 = 0 ...(ii)

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

By (i) x 3 + (ii) x 4, we get

25x = 68

∴ x = `68/25`

Substituting x = `68/25` in (ii), we get

`4(68/25) + 3y - 5` = 0

∴ 3y = `5 - 4(68/25) = (125 - 272)/25 = (-147)/25`

∴ y = `(-49)/25`

∴ The co-ordinates of the foot of perpendicular M are `(68/25, (-49)/25)`.