• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Shortest Distance

  • Thread starter Air
  • Start date

Air

202
0
1. Homework Statement
Find the shortest distance from the origin to the curve [tex]x^2+2xy+y^2=150[/tex].


2. Homework Equations
[tex]\frac{\partial f}{\partial x}[/tex], [tex]\frac{\partial f}{\partial y}[/tex]


3. The problem I'm occuring
I'm not sure how to start is thus can't attempt it. I would have used the Lagrange theory but that would give the max point and I also don't have the constraint so cannot use it. For a start, can someone suggest how I would start. Any methods? I have to use partial differentiation.
 
316
0
You can use Lagrange's multiplier method to find maxima as well as minima. Apply it to the distance squared function f(x,y)=x^2+y^2 with the constraint x^2+2xy+y^2=150.
 

Air

202
0
I have reached a point of confusion...

[tex]f(x,y) = x^2+y^2[/tex]
[tex]g(x,y) = x^2+2xy+y^2=150[/tex]

[tex]\mathrm{Equation 1: }2x - \lambda (2x + 2y) = 0 \implies \lambda = \frac{2x}{2x+2y}[/tex]
[tex]\mathrm{Equation 2: }2y - \lambda (2x + 2y) = 0 \rightarrow 2y - \left( \frac{2x}{2x+2y}\right) (2x + 2y ) = 0 \rightarrow 2x+2y = 0 \implies y=x[/tex]

Into constraint equation to solve for value of [tex]x[/tex] and [tex]y[/tex].
[tex]\therefore y^2 + 2y^2 y^2 = 150 \implies y = \pm \sqrt{\frac{150}{4}} = x[/tex]

Which gives many solutions. How can I saw which is the shortest distance?
 
316
0
Well, you got two solutions: [tex](\sqrt{150}/2,\sqrt{150}/2)[/tex] and [tex](-\sqrt{150}/2,-\sqrt{150}/2)[/tex]. Both have the same distance from the origin, compute it and you have the answer.
 

Related Threads for: Shortest Distance

  • Posted
Replies
10
Views
4K
  • Posted
Replies
5
Views
3K
  • Posted
Replies
1
Views
3K
  • Posted
Replies
0
Views
3K
Replies
8
Views
5K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top