<s>[INST] Find largest possible value of $x^2+y^2$ given that $x^2+y^2=2x-2y+2$ 
Let $x, y \in \mathbb R$ such that $x^2+y^2=2x-2y+2$.  Find the largest possible value of $x^2+y^2$.

My attempt:
$x^2+y^2=2x-2y+2$
$(x^2-2x)+(y^2+1)=2$
$(x-1)^2+(y+1)^2=4$
I have no idea how to continue here.  Any help?
 [/INST] You may solve this problem using geometry or you may use Lagrange Multipliers method. 
You have already found $$(x-1)^2+(y+1)^2=4$$ and you want to find a point on this circle which maximizes $$x^2+y^2$$. 
Graphing the first circle and finding the largest possible circle  $$x^2+y^2= R^2$$ to have a point on the circle $$  (x-1)^2+(y+1)^2=4$$ leads to the point with $ x=1+\sqrt 2$ and $y=-1-\sqrt 2$.
That implies $x^2= y^2 =3+2\sqrt 2 $ which in  turns gives   $$x^2+y^2= 2(3+2\sqrt 2) =11.6568.. $$
The Lagrange Multipliers Methods provides the same results.
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