Find the point on the line y = 2x + 3 that is closest to the origin.

Solution:

Given, equation of the line is y = 2x + 3 ---------(1)

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

We need to find an equation of the perpendicular from (0,0) on y = 2x + 3.

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,

5y = 3 

y = 3/5 

x + 2(3/5) = 0

x = -6/5

x = -6/5 and y = 3/5

Therefore, the point on the line is (-6/5, 3/5).

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Find the point on the line y = 2x + 3 that is closest to the origin.

Summary:

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