<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] $x^2 + y^2$ is essentially the square of the distance of a point $(x, y)$ from the origin. Now, the question is asking us to find two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ which lie on the circle and are at a minimum and maximum distance from the origin.
It makes sense that these two points will lie on the line joining the center of the circle and the origin. I won't be proving this here, but it should be pretty obvious upon drawing a rough diagram.
For the given circle
$$
x^2 - 4x + y^2 + 3 = 0
$$
the center is $(2,0)$. Hence, we draw a line passing through this point and the origin (the red line in the below diagram).

The red line is
$$
y = 0
$$
Now we find the point of intersection between this line and the circle.
$$
x^2 - 4x + 3 = 0 \\
(x - 3)(x - 1) = 0 \\
x = 1, 3
$$
Since $y = 0$, the maximum and minimum value of $x^2 + y^2$ is $1$ and $9$.
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