Lagrange on an Ellipse to find Max/Min Distance

[YayMathYay]

Homework Statement

Homework Equations

Lagrance Multipliers.

The Attempt at a Solution

This is a pretty dumb question, and I feel a little embarassed asking but..

I know how to do the Lagrange part (I think). I'm assuming you maximize/minimize the distance, $\sqrt{x^{2} + y^{2} + z^{2}}$ subject to the constraints of the ellipse.


The part I'm having trouble with is finding the constraints of the ellipse - could someone help point me in the right direction?

 

[Ray Vickson]

Homework Statement

Homework Equations

Lagrance Multipliers.

The Attempt at a Solution

This is a pretty dumb question, and I feel a little embarassed asking but..

I know how to do the Lagrange part (I think). I'm assuming you maximize/minimize the distance, $\sqrt{x^{2} + y^{2} + z^{2}}$ subject to the constraints of the ellipse.


The part I'm having trouble with is finding the constraints of the ellipse - could someone help point me in the right direction?

It is a lot easier to maximize or minimize distance-squared; do you see why it is OK to do that? As for the constraints, ask yourself: how can I tell if a given point (x,y,z) is on the ellipse? What are the conditions for that?

 

[YayMathYay]

It is a lot easier to maximize or minimize distance-squared; do you see why it is OK to do that? As for the constraints, ask yourself: how can I tell if a given point (x,y,z) is on the ellipse? What are the conditions for that?

Ah yes, the thought of it passed my mind, but I didn't think it was viable until you pointed it out - thanks! :)

As for the constraints, I know the equations of the cylinder and the plane that create it.. but when I try to put them together (i.e., $x^{2} + y^{2} = x + y + z$), it doesn't seem to give me the correct equation. I might just be having a brain fart, but I just don't know what I'm missing or doing incorrectly here :(

 

[YayMathYay]

Just gave it an attempt, and my maximization problem became:Maximize $x^{2} + y^{2} + z^{2}$ subject to $x + y - \sqrt{1 - x^{2}} - \sqrt{1 - y^{2}} = 0$

Is this correct? If so, do I need to substitute out z in terms of x and y? Because I have a feeling that would make things very complicated..

 

[Ray Vickson]

Just gave it an attempt, and my maximization problem became:


Maximize $x^{2} + y^{2} + z^{2}$ subject to $x + y - \sqrt{1 - x^{2}} - \sqrt{1 - y^{2}} = 0$

Is this correct? If so, do I need to substitute out z in terms of x and y? Because I have a feeling that would make things very complicated..

You wrote two equations, not one, so you need two Lagrange multipliers.