I've been working on a problem for a long time now, and I'm a little bit stuck on setting up the problem, and I just need a little bit of advice or explanation on how to work on the problem.
So the problem is essentially that you begin at some point (4, -4) and you need to get to another point (7,9). You want to minimize this distance using Lagrangian. Up to here, it's a fairly standard problem, but there is a caveat. You want to stop by a "table" which is essentially an ellipse in the middle of the coordinate system first.
My basic approach to it was the following:
You want to minimize your distance, so your f function is just distance formula with the first point being where you begin at (4,-4) so
$$f(x,y) = (x-4)^2 +(y+4)^2$$
{I didn't square root the function because the minimum distance should have the same solution as the minimum distance squared}
Then I started writing my constraints.
As g(x,y) because you have to go to the table, I figured the constraint was the following:
$$g(x,y) = x^2/9 +y^2/1 $$
which is just the equation of the ellipse "table" in the center. I'm really confused as to if what I have is correct, and also what to choose as my c values.
Thank you in advance for your help! I'm also fairly new so please help me ask better questions if you think anything needs clarifying.
Essentially, this problem is asking you to find the point on the ellipse such that the sum of the straight-line distances from it to the two fixed points. So, the constraint is indeed the equation of the ellipse, but the objective function is going to be a sum of two square roots: the distance from $(x,y)$ to $(4,-4)$ and the distance from $(x,y)$ to $(7,9)$. You probably won’t be able to use the usual trick of using the square of the distance to simplify the equations.
This formulation includes paths that go through the table, but those are all longer than ones that don’t, so if you’ve done this correctly (and whoever posed the problem set it up well), you won’t get a minimal path that does this.