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.