Shortest & largest distance from origo to ellipse

[Inertigratus]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

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

 

[HallsofIvy]

If x^2= 2 then x= \pm\sqrt{2}. Put that back into the equation of the ellipse and solve for y.

 

[Inertigratus]

Yes, but I got x^2 = 2 from that equation, only because x = y.
I meant the other optima. According to the answers, 2 is the minimum. So I'm looking for the maximum.

 

[Inertigratus]

Ohh, nevermind... now that I checked the answer, x = -y.
The thing is, I got x^2 = y^2 and assumed that x = y, but obviously x = -y is correct too and gives different points.
Thanks anyway :).