find the points on the line y=10x+9 that is closest to (0,0).

A) Draw a picture
B)imagine an arbitrary Point on the line (x,y) and y=10x+9
C)Find a function in one variable which measures the distance from P to (0,0)
D) Find the minimum value of the function
E) What is the point we are looking for?

I can't draw a picture.
The line has slope 10 and intersect the y axis at x=9.
A picture would say it is somewhere around x = -1/2.

Minimizing the distance will gives the same result as minimizing the square of the distance.
The distance squared from the origin to the graph would be x+y.
Putting in y = 10x+9 gives x + (10x+9).

That is the same as 101x + 180x + 81.

The derivative of this is 202x + 180.
Setting that to 0 gives 202x = -180.
That means the minimum is at x = -90/101.

Checking this out gives
-92/101   0.841584158
-91/101   0.811881188
-90/101   0.801980198
-89/101   0.811881188
-88/101   0.841584158

As can be seen, x = -90/101 gives the closest.


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