General Equation of Second Degree:
The equation of the form
Where and are not simultaneous 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
(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 is vanished in the new coordinates axes.
Example: Identify the conic represented by the equation .
Compare with the general equation of conic we have , and , so
This shows that the given equation represents the ellipse.