# How do you classify the conic x^2+y^2-2x+6y+9=0?

Nov 30, 2016

We have a a circle, center $\left(1 , - 3\right)$ and radius $= 1$

#### Explanation:

The general equation of a conic is

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

We calculate the discriminant $\Delta = {B}^{2} - 4 A C$

Here, we have

${x}^{2} + {y}^{2} - 2 x + 6 y + 9 = 0$

$\Delta = 0 - 4 \cdot 1 \cdot 1 = - 4$

$\Delta < 0$, so we have a circle, center $\left(1 , - 3\right)$ and radius $= 1$

${x}^{2} - 2 x + {y}^{2} + 6 y = - 9$

${x}^{2} - 2 x + 1 + {y}^{2} + 6 y + 9 = - 9 + 1 + 9 = 1$

${\left(x - 1\right)}^{2} + {\left(y + 3\right)}^{2} = 1$

graph{x^2+y^2-2x+6y+9=0 [-5.2, 4.662, -4.088, 0.845]}