Maximum and minimum of a function of 2 variables (Lagrange multipliers)

My question is : **Can I solve the problem without passing by Lagrange multipliers?** Because I have some difficulties by doing so. **If not, then I don't want any tip and I'll work harder.**

I must find the minimum and the maximum of the function in the ellipse .

So it seems a problem that can be solved via Lagrange multipliers, but I didn't find a way to find , and . (I even had to divide by if but then I found another condition ( if then ...) and even with it I couldn't isolate nor nor ).