<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] Although this kind of problem is usually solved using Lagrange multipliers, this one can be solved using methods from precalculus.
We wish to maximize the value of $x^2+y^2$ on the circle
\begin{equation}
(x-2)^2+y^2=1
\end{equation}
Which is defined on the interval $[1,3]$.
Since $x^2+y^2=4x-3$ the maximum and minimum values of $x^2+y^2$ will be the maximum and minimum values of $4x-3$ on the interval $[1,3]$, namely $1$ and $9$.
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