Finding extremal distances from origin to ellipse 13x² + 13y² + 10xy = 72

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Homework Statement


Find the largest and shortest distance from origo to the ellipse.

Homework Equations


Ellipse: [itex]g(x, y) = 13x^2 + 13y^2 + 10xy = 72[/itex]
Function to optimize: [itex]F(x, y) = \sqrt{x^2 + y^2}[/itex]
But this is easier to optimize: [itex]f(x, y) = x^2 + y^2[/itex]

The Attempt at a Solution


I set up the equations, [itex]\nabla f = \lambda \nabla g[/itex], which got me that [itex]x = y[/itex]. [itex]g(x) = 36x^2 = 72[/itex] and [itex]x^2 = 2[/itex] which got me that one of the values (either minimum or maximum) is [itex]F(x, y) = 2[/itex].

The question is, how do I get the other value?
 
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Discussion

If [itex]x^2= 2[/itex] then [itex]x= \pm\sqrt{2}[/itex]. Put that back into the equation of the ellipse and solve for y.
 
Yes, but I got [itex]x^2 = 2[/itex] from that equation, only because [itex]x = y[/itex].
I meant the other optima. According to the answers, 2 is the minimum. So I'm looking for the maximum.
 
Ohh, nevermind... now that I checked the answer, [itex]x = -y[/itex].
The thing is, I got [itex]x^2 = y^2[/itex] and assumed that [itex]x = y[/itex], but obviously [itex]x = -y[/itex] is correct too and gives different points.
Thanks anyway :).