The question: Find the maximum distance from the origin to the ellipse $\displaystyle x^2+xy+y^2=3.3$.

My work: I figure that the distance to the origin is defined by the equation $\displaystyle \sqrt{x^2+y^2}$ so that is the function I need to maximize subject to the constraint defined by the equation of the ellipse.

So I've formed the Lagrangian $\displaystyle L= \sqrt{x^2+y^2} + \mu (x^2+xy+y^2-3.3)$ and taken the partials:

$\displaystyle \frac{\partial L}{\partial x}= \frac{2x}{2 \sqrt{x^2+y^2}}-2\mu x-\mu y $

$\displaystyle \frac{\partial L}{\partial y}= \frac{2y}{2 \sqrt{x^2+y^2}}-2\mu y-\mu x $

$\displaystyle \frac{\partial L}{\partial \mu}= -x^2-xy-y^2+3.3$

After setting all the partials equal to zero, I don't know where to go. No obvious algebraic solutions are coming to mind.