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

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Answer 1 · 2016-10-02T06:58:29

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

Let the equation be of the type $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

then if

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

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

$B^2-4AC<0$ and $A!=C$ , it is an ellipse

$B^2-4AC>0$ , it is a hyperbola

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

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

Therefore, $B^2-4AC=0^2-4xx1xx0=-0$ and $C=0$ but $A!=0$

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

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