Find the point on the line y = 2x + 4 that is closest to the origin.

Solution:

Given, equation of the line is y = 2x + 4 --- (1)

Closest point from origin will be the perpendicular distance from origin to the line.

We need to find out equation of the perpendicular from (0,0) on y = 2x + 4.

The equation is in slope intercept form i.e. y = mx + c

Slope, m = 2

Slope of the perpendicular = -(1/m) = -1/2

Equation of the perpendicular is found by (y - y₁) = m (x - x₁)

⇒ y - 0 = (-1/2) (x - 0)

⇒ y = (-1/2)x

⇒ 2y + x = 0 --- (2)

Solving (1) and (2), we get,

⇒ 5x = -8

⇒ x = -8/5

Substiute x = -8/5 in eq(2), we get

⇒ y = 4/5

x = -1.6 and y = 0.8

Therefore, the point on the line y = 2x + 4 closest to origin is (-1.6, 0.8)

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Find the point on the line y = 2x + 4 that is closest to the origin.

Summary:

The point on the line y = 2x + 4 that is closest to the origin is (-1.6 , 0.8).