# How do you graph  x^2+y^2-2x+6y+6=0?

Feb 10, 2016

Re-write the equation in standard circle format;
then draw a circle with the center and radius given by the equation.

#### Explanation:

Given
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{x}^{2}} + \textcolor{b l u e}{{y}^{2}} \textcolor{red}{- 2 x} \textcolor{b l u e}{+ 6 y} \textcolor{g r e e n}{+ 6} = 0$

Rearrange grouping the $x$ terms, the $y$ terms, and with the constant on the right side
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{x}^{2} - 2 x} + \textcolor{b l u e}{{y}^{2} + 6 y} = \textcolor{g r e e n}{- 6}$

Complete the square for the $x$ sub-expression and the $y$ sub-expression:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{x}^{2} - 2 x + 1} + \textcolor{b l u e}{{y}^{2} + 6 y + 9} = \textcolor{g r e e n}{- 6} \textcolor{red}{+ 1} \textcolor{b l u e}{+ 9}$

Write as the sum of squared binomials equal to a square
color(white)("XXX")color(red)((x-1)^2)+color(blue)((y+3)^2=color(green)(2^2)

Note that the equation for a circle with center $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$ and radius $\textcolor{g r e e n}{r}$ is
$\textcolor{w h i t e}{\text{XXX}} {\left(x - \textcolor{red}{a}\right)}^{2} + {\left(y - \textcolor{b l u e}{b}\right)}^{2} = {\textcolor{g r e e n}{r}}^{2}$

So we need to draw a circle with center at $\left(\textcolor{red}{1} , \textcolor{b l u e}{- 3}\right)$ and radius $\textcolor{g r e e n}{2}$

graph{x^2+y^2-2x+6y+6=0 [-5.354, 4.51, -4.946, -0.015]}