# General Equation of a Conic

__General Equation of the Second Degree__**:**

The equation of the form

where and are not simultaneously zero is called the general equation of the second degree or the quadratic equation in and .

Now we state the following theorem which indicates that the general second degree equation represents the general equation of conics, and the classification of conics depends on the constant .

__Theorem__**:** The general equation of the second degree represents a conic section. It represents a:

**(i) ** Parabola if

**(ii)** Ellipse if

**(iii)** Hyperbola if

__Theorem__**:** If the axes are rotated about the origin through an angle , where is given by , then the product terms in the general second degree equation vanish in the new coordinates' axes.

__Example__**:** Identify the conic represented by the equation .

Comparing with the general equation of a conic we have , and , so

This shows that the given equation represents an ellipse.