# Shortest & largest distance from origo to ellipse

1. Aug 13, 2011

### Inertigratus

1. The problem statement, all variables and given/known data
Find the largest and shortest distance from origo to the ellipse.

2. Relevant 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: $f(x, y) = x^2 + y^2$

3. 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?

2. Aug 13, 2011

### HallsofIvy

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

3. Aug 13, 2011

### 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.

4. Aug 13, 2011

### 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 :).