Find the point on the line y = 3x + 4 that is closest to the origin?

Solution:

The point on line y = 3x + 4 closest to the origin is the point of intersection of y = 3x +4 and line perpendicular to it passing through (0, 0).

We know that equation of line perpendicular to Ax + By + C = 0 and passing through (x₁, y₁) is

⇒ B(x - x₁) - A(y - y₁) = 0.

Here, A = 3, B = -1 and (x₁, y₁) = (0, 0)

⇒ -1(x - 0) - 3(y - 0) = 0

⇒ x + 3y = 0 --- (1)

Now, the nearest point, say P, on line y = 3x + 4 from the origin is the point of intersection of 3x - y + 4 = 0 and x + 3y = 0.

Solving, x + 3(3x + 4) = 0

⇒ 10x + 12 = 0

⇒ x = -12/10

⇒ x = -6/5

Substituting x = -6/5 in equation (1), we get

⇒ -6/5 + 3y = 0

⇒ y = 2/5

The required point is (-6/5, 2/5)

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Find the point on the line y = 3x + 4 that is closest to the origin?

Summary:

The point on the line y = 3x + 4 that is closest to the origin is (-6/5, 2/5).