Find the equation of the perpendicular dropped from the point (-1, 2) onto the line joining the points (1, 4) and (2, 3).

Let P = (-1, 2)

Let A and B be the points (1, 4) and (2, 3).

$\text{Slope of AB }= \dfrac{3 - 4}{2 - 1} \\[1em] = \dfrac{-1}{1} = -1.$

We know that,

Product of slope of perpendicular lines is -1.

Let slope of line through P be m.

∴ m × Slope of AB = -1

⇒ m × -1 = -1

⇒ -m = -1

⇒ m = 1.

By point-slope form,

Equation of line through P,

⇒ y - y₁ = m(x - x₁)

⇒ y - 2 = 1[x - (-1)]

⇒ y - 2 = 1(x + 1)

⇒ y - 2 = x + 1

⇒ y - x = 1 + 2

⇒ y - x = 3

⇒ y = x + 3

Hence, equation of the perpendicular dropped from the point (-1, 2) onto the line joining the points (1, 4) and (2, 3) is y = x + 3.