# How do you identify the type of conic #4x^2+8y^2-8x-24=4# is, if any and if the equation does represent a conic, state its vertex or center?

##### 1 Answer

An ellipse

#### Explanation:

Conics can be represented as

where

For conics

The characteristic polynomial is

Depending on their roots, the conic can be classified as

1) Equal --- circle

2) Same sign and different absolute values --- ellipse

3) Signs different --- hyperbola

4) One null root --- parabola

In the present case we have

with characteristic polynomial

with roots

Being an ellipse there is a canonical representation for it

giving

solving we get

so