Find the point on the line y = 3x + 5 that is closest to the origin.

Solution:

The point on line y = 3x + 5 closest to the origin is the point of intersection of y = 3x + 5 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 + 5 from the origin is the point of intersection of 3x - y + 5 = 0 and x + 3y = 0.

Solving, x + 3(3x + 5) = 0

⇒ 10x = -15

⇒ x = -15/10

⇒ x = -3/2

Substituting the x = -3/2 in equation(1)

⇒ -3/2 + 3y = 0

⇒ y = 1/2

Therefore, the required point is (-3/2, 1/2)

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Find the point on the line y = 3x + 5 that is closest to the origin.

Summary:

The point on the line y = 3x + 5 that is closest to the origin is (-3/2, 1/2).