<s>[INST] completing square for a circle In the following question: 

I don't understand how we can get from the original equation to the final equation using completing the square.
Any thoughts as how to get to the final equation?
 [/INST] One way of doing this is to avoid fractions completely until the last step.
$$\begin{align}
\frac{x+y}{x^2+y^2+1}&=c\\
x+y&=c(x^2+y^2+1)\\
x+y&=cx^2+cy^2+c\\
cx^2-x+cy^2-y+c&=0\\
cx^2-x+cy^2-y&=-c\\
4c(cx^2-x+cy^2-y)&=4c(-c)\\
4c^2x^2-4cx+4c^2y^2-4cy&=-4c^2\\
(2cx)^2-2(2cx)+(2cy)^2-2(2cy)&=-4c^2\\
(2cx)^2-2(2cx)+1+(2cy)^2-2(2cy)+1&=2-4c^2\\
(2cx-1)^2 + (2cy-1)^2 &= 2-4c^2\\
\frac{(2cx-1)^2 + (2cy-1)^2}{4c^2} &= \frac{2-4c^2}{4c^2}\\
\bigg(x-\frac{1}{2c}\bigg)^2 + \bigg(y-\frac{1}{2c}\bigg)^2 &= \frac{1}{2c^2}-1\\
\end{align}$$
</s>