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

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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.

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