Minimizing Distance to Origin: Solving an Optimization Problem

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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?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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