Question

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

Answer 1

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

Explanation:

Given
`color(white)("XXX")color(red)(x^2)+color(blue)(y^2)color(red)(-2x)color(blue)(+6y)color(green)(+6)=0`

Rearrange grouping the `x` terms, the `y` terms, and with the constant on the right side
`color(white)("XXX")color(red)(x^2-2x)+color(blue)(y^2+6y)=color(green)(-6)`

Complete the square for the `x` sub-expression and the `y` sub-expression:
`color(white)("XXX")color(red)(x^2-2x+1)+color(blue)(y^2+6y+9) = color(green)(-6)color(red)(+1)color(blue)(+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 `(color(red)(a),color(blue)(b))` and radius `color(green)(r)` is
`color(white)("XXX")(x-color(red)(a))^2+(y-color(blue)(b))^2=color(green)(r)^2`

So we need to draw a circle with center at `(color(red)(1),color(blue)(-3))` and radius `color(green)(2)`

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