# How do you identity if the equation 3x^2+4y^2+8y=8 is a parabola, circle, ellipse, or hyperbola and how do you graph it?

Nov 19, 2016

This is an ellipse

#### Explanation:

Compare this to the general equation for conics and calculate the discriminant.

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

$3 {x}^{2} + 4 {y}^{2} + 8 y - 8 = 0$

The discriminant is

$\Delta = {B}^{2} - 4 A C$

if $\Delta < 0$, we have an ellipse

If $\Delta = 0$, we have a parabola

If $\Delta > 0$, we have a hyperbola

In our case,

$\Delta = 0 - 4 \cdot 3 \cdot 4 = - 48$

so, $\Delta < 0$, we have an ellipse.

We can also, rearrage the equation

$3 {x}^{2} + 4 \left({y}^{2} + 2 y\right) = 8$

$3 {x}^{2} + 4 \left({y}^{2} + 2 y + 1\right) = 8 + 4$

$3 {x}^{2} + 4 {\left(y + 1\right)}^{2} = 12$

Dividing by 12

${x}^{2} / 4 + {\left(y + 1\right)}^{2} / 3 = 1$

which is an equation of an ellipse

The center is $\left(0 , - 1\right)$

$c = \sqrt{4 - 3} = 1$

The foci are F $= \left(1 , - 1\right)$ and F'$= \left(- 1 , - 1\right)$

The major axis is $= 4$

and the minor axis is $= \sqrt{3}$

graph{(3x^2+4y^2+8y-8)(y-1)=0 [-4.915, 3.855, -3.362, 1.023]}