Math Problem Statement
- Find the point on the line y = 2x + 3 that is closest to the origin.
Solution
To find the point on the line that is closest to the origin, we need to minimize the distance from a point on the line to the origin . The distance from the origin to a point is given by:
Since , we can substitute for in the distance formula:
To minimize , it suffices to minimize because the square root is a monotonically increasing function.
Expand :
Thus, becomes:
To find the minimum, take the derivative of with respect to and set it to zero:
Set the derivative equal to zero:
Solve for :
x = -\frac{6}{5}$$ Substitute $$x = -\frac{6}{5}$$ back into the line equation to find $$y$$: $$y = 2\left(-\frac{6}{5}\right) + 3 = -\frac{12}{5} + \frac{15}{5} = \frac{3}{5}$$ Therefore, the point on the line $$y = 2x + 3$$ that is closest to the origin is: $$\left(-\frac{6}{5}, \frac{3}{5}\right)$$Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Distance Formula
Optimization
Formulas
Distance formula: d = sqrt(x^2 + y^2)
Derivative to find minimum: f'(x) = 0
Theorems
Monotonicity of square root function
Optimization principles
Suitable Grade Level
Grades 10-12