# How do you classify the conic 3x^2+y^2+2x+2y=0?

Jan 20, 2017

#### Answer:

$3 {x}^{2} + {y}^{2} + 2 x + 2 y = 0$ is an ellipse.

#### 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 $3 {x}^{2} + {y}^{2} + 2 x + 2 y = 0$

$A = 3$, $B = 0$ and $C = 1$

Therefore, ${B}^{2} - 4 A C = {0}^{2} - 4 \times 3 \times 1 = - 12 < 0$ and $A \ne C$

Hence, $3 {x}^{2} + {y}^{2} + 2 x + 2 y = 0$ is an ellipse.
graph{3x^2+y^2+2x+2y=0 [-2.55, 2.45, -2.2, 0.3]}