Optimization Part IV

[Hckyplayer8]

Find the shortest (straight line) distance from the positive x axis to the positive y axis, through point (8,1).

I have no idea where to start on this one. I understand what the problem is asking. However, this is a challenge practice question where no example was offered.

 

[Deleted member 4993]

Find the shortest (straight line) distance from the positive x axis to the positive y axis, through point (8,1).

I have no idea where to start on this one. I understand what the problem is asking. However, this is a challenge practice question where no example was offered.

Start with assuming that the shortest line intersects y-axis at (0,y₁) and it intersects the x-axis at (x₁,0)

Make sure it passes through (8,1)

Now minimize the distance between (x₁,0) and (0,y₁). It might be helpful if you draw a sketch.

 

[Steven G]

Here is a slighty different approach.
Draw a sketch!
What is the general equation of this line that passes through (8,1) and is not vertical or horizontal? Why do we not want vertical or horizontal?
What is the distance of this line from the x-intercept to the y-int? Then minize this distance?

Think about the conection between the slope of this line and the two intercepts

 

[Hckyplayer8]

Start with assuming that the shortest line intersects y-axis at (0,y₁) and it intersects the x-axis at (x₁,0)

Make sure it passes through (8,1)

Now minimize the distance between (x₁,0) and (0,y₁). It might be helpful if you draw a sketch.

Here is a slighty different approach.
Draw a sketch!
What is the general equation of this line that passes through (8,1) and is not vertical or horizontal? Why do we not want vertical or horizontal?
What is the distance of this line from the x-intercept to the y-int? Then minize this distance?

Think about the conection between the slope of this line and the two intercepts

I sketched the positive area of the x and y axis and placed my point at x = 8 and y =1. We do not want a vertical or horizontal line because the line must intersect both the x and y axis.

Just eyeing the setup, as the y intercept approaches 1 (which it can't be or otherwise we have a horizontal line), the x intercept is moved farther and farther positively down the axis. The same goes for the y intercept as the x intercept approaches 8. The y intercept moves positively up the y axis.

So there is going to be that "sweet spot" of minimal distance.

Since we don't know the y intercept, slope intercept form is useless.

Point slope form would be y-1 = m(x-8)

 

[Steven G]

I don't exactly agree with you that we do not know the y intercept but you are partly correct. After all, armed with the y-int and the point (8,1) we would know the equation of the line.

More importantly what is the equation for the distance between the two intercepts. We must have this equation so that we can minimize it!

 

[Deleted member 4993]

The equation of the line joining (x₁,0) & (0,y₁) is:

x/x₁ + y/y₁ = 1

This line goes through (8,1). Hence:

8/x₁ + 1/y₁ = 1

Assume that the length of the line joining (x₁,0) & (0,y₁) is D. Then:

D² = [x₁]² + [y₁]²

Now what can you do with these information........