# How do you classify the conic x^2+6x-2y+13=0?

Oct 2, 2016

${x}^{2} + 6 x - 2 y + 13 = 0$ is a parabola.

#### Explanation:

Let the equation be of the type $A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

then if

${B}^{2} - 4 A C = 0$ and $A = 0$ or $C = 0$, it is a parabola

${B}^{2} - 4 A C < 0$ and $A = C$, it is a circle

${B}^{2} - 4 A C < 0$ and $A \ne C$, it is an ellipse

${B}^{2} - 4 A C > 0$, it is a hyperbola

In the given equation ${x}^{2} + 6 x - 2 y + 13 = 0$

$A = 1$, $B = 0$ and $C = 0$

Therefore, ${B}^{2} - 4 A C = {0}^{2} - 4 \times 1 \times 0 = - 0$ and $C = 0$ but $A \ne 0$

Hence, ${x}^{2} + 6 x - 2 y + 13 = 0$ is a parabola.
graph{x^2+6x-2y+13=0 [-13.79, 6.21, -0.92, 9.08]}