Minimizing Distance to Origin: Solving an Optimization Problem

[Anabelle37]

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!

 

[micromass]

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...

 

[Mark44]

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

 

[Anabelle37]

thanks :)

 

[HallsofIvy]

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?