<s>[INST] Maximize $k=x^2+y^2$ Subject to $x^2-4x+y^2+3=0$ Question 
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. Find the maximum and minimum values of $x^2+y^2$. 
My work
Let $k=x^2+y^2$
Therefore, $x^2-4x+y^2+3=0$ ---> $k-4x+3=0$ .
What do I do next? How do I find an expression in terms of $k$ that I can maximize?
 [/INST] $F(x,y,\lambda)=x^2+y^2+\lambda (x^2-4 x+y^2+3)$ is lagrange multiplier. We need to solve system of equations $$F'_x=0$$ $$F'_y=0,$$ $$x^2-4 x+y^2+3=0$$ or $$2x+2\lambda x-4\lambda=0,$$ $$2y+2\lambda y=0,$$ $$x^2-4 x+y^2+3=0.$$ We have $(x,y,\lambda)\in\{(1,0,1),(3,0,-3)\}$. For $\lambda=1$ is $d^2 F>0$ so $(x,y)=(1,0)$ is point of minimum and $(x,y)=(3,0)$ is point of maximum, since $d^2 F<0$ for $\lambda=-3$. 
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