How do I find this distance?

Question

Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2 = 1$ and the origin.

This is what I've attempted so far:

Maximize $x^2+y^2+z^2$ with respect to $x^2+xy+2y^2 = 1$. Using Lagrange multipliers, we get the system of equations:

$$2x = \lambda(2x+y)$$ $$2y = \lambda(x+4y)$$ $$0 = \lambda z$$ $$x^2+xy+2y^2 = 1$$

I do not know how to solve this system. Can someone please help me out?

Answer 1

Hint: You can minimize/maximize $r^2=x^2+y^2$ instead of $r$. You have $r^2=-y^2-xy+1$ Use your equation to get an expression for $x$ in terms of $y$ and substitute it in. Take the derivative with respect to $y$, set to zero, etc.

Answer 2

I would suggest finding the axes of the ellipse using purely geometry (plus coordinate algebra) ways. This may require some knowledge of how to manipulate bilinear forms but I think most precalc textbooks cover it for conics, so I assume fair game.