Find the foot of the perpendicular from the point (3, 8) to the line x

1.	Find the foot of the perpendicular from the point (3, 8) to the line x + 3y = 7.
Answer» The given equation of the line is x + 3y = 7 ⇒ y = − \(\frac{1}{3}\)x + \(\frac{7}{3}\) ∴ Slope of the line, m₁ = – \(\frac{1}{3}\) Let m₂ be the slope of the Perpendicular. ∴ m₁ m₂ = – 1 ⇒ − \(\frac{1}{3}\) × m₂ = − 1 ⇒ m₂ = 3. ∴ Equation of the perpendicular line with slope 3 and passing through (3, 8) is y – 8 = 3 (x – 3) ⇒ 3x – y – 1 = 0 ∴ The foot of the perpendicular is the point intersection of the lines x + 3y – 7 = 0 and 3x – y – 1 = 0 Solving these equations, we get x = 1, y = 2, so (1, 2) is the foot of the perpendicular.

Find the foot of the perpendicular from the point (3, 8) to the line x + 3y = 7.

Answer»

The given equation of the line is x + 3y = 7

⇒ y = − \(\frac{1}{3}\)x + \(\frac{7}{3}\)

∴ Slope of the line, m₁ = – \(\frac{1}{3}\)

Let m₂ be the slope of the Perpendicular.

∴ m₁ m₂ = – 1

⇒ − \(\frac{1}{3}\) × m₂ = − 1

⇒ m₂ = 3.

∴ Equation of the perpendicular line with slope 3 and passing through (3, 8) is

y – 8 = 3 (x – 3)

⇒ 3x – y – 1 = 0

∴ The foot of the perpendicular is the point intersection of the lines x + 3y – 7 = 0 and 3x – y – 1 = 0

Solving these equations, we get

x = 1, y = 2, so (1, 2) is the foot of the perpendicular.