You could do this without Lagrange by solving the ellipse equation for y and subbing into , differentiating, setting to 0 and solving for x.
It will take some algebra, but it certainly doable.
The question: Find the maximum distance from the origin to the ellipse .
My work: I figure that the distance to the origin is defined by the equation 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 and taken the partials:
After setting all the partials equal to zero, I don't know where to go. No obvious algebraic solutions are coming to mind.