<s>[INST] Plotting  a quadratic equation in the $\,xy\,$- plane My question is:
Represent the following set of points in the $\,xy\,$- plane:
$$\left\{ (x,y)\,\, |\,\, x^2 + y^2 - 2x - 2y + 1 = 0 \right\}$$
What i got: $\,\,(x-2)^2 + (y-2)^2 = 1\,\,$
I am not getting what to do next.
Any help to solve this question would be greatly appreciated.
Thank you,
 [/INST] When given an equation of the form $x^2-2x+y^2-2y+1=0$ the first step is to complete the square for $x$ and for $y$.
The idea is that if we have $x^2-2x$ we can write it as $(x+C)^2+D$ instead. Since know those that the coefficient of $x$ is $2C$, we know that $C=-1$, so we have: $$(x-1)^2=x^2-2x+1\implies x^2-2x = (x-1)^2-1$$
Therefore we can write it as $(x-1)^2-1$.
Similarly we can replace $y^2-2y$ by a similar term, so we have now:
$$\begin{align}
&\underbrace{x^2-2x}+\underline{y^2-2y}+1 = 0 &&\text{complete the squares}\\
&\underbrace{(x-1)^2-1}+\underline{(y-1)^2-1}+1 =0 &&\text{sum the }1\text{'s}\\
&(x-1)^2+(y-1)^2-1=0 &&+1\\
&(x-1)^2+(y-1)^2=1 &&\text{ circle!}
\end{align}$$
Therefore we have a circle of radius $1$ whose center is at $(1,1)$.
</s>