Minimizing Distance to Origin: Solving an Optimization Problem

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Homework Help Overview

The problem involves finding the point on the line defined by the equation y=4x+7 that is closest to the origin, which falls under the subject area of optimization in mathematics.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss different methods to approach the problem, including minimizing the distance to the origin and considering the square of the distance to avoid square roots. There is also mention of a geometric interpretation involving perpendicular lines.

Discussion Status

The discussion is active with various approaches being explored. Some participants have suggested methods for minimizing distance, while others are considering geometric interpretations. There is no explicit consensus on a single approach yet.

Contextual Notes

The original poster expresses confusion about how to begin the problem, indicating a need for clarification on the concepts involved in optimization.

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URGENT - optimisation problem

Homework Statement



Find the point on the line y=4x+7 that is closest to the origin.

Homework Equations





The Attempt at a Solution



I am completely lost on where to start and what to minimise. It's for exam preparation!
 

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Take an arbitrary point on the line. This has the form (x,4x+7). What is the distance to the origin? You will have to minimize that...
 


Or you can minimize d2, the square of the distance, which doesn't involve a square root.
 


thanks :)
 


Or you can do it completely geometrically- the shortest distance from a point to a line is along the line, through the point, perpendicular to the given line.

y= 4x+ 7 has slope 4. A line perpendicular to it will have slope -1/4. What is the equation of a line, through (0, 0), with slope -1/4? Where does that line intersect y= 4x+ 7?
 

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