General Equation of a Conic

General Equation of the Second Degree:

The equation of the form

a{x^2} = b{y^2} + 2hxy + 2gx + 2fy + c = 0

where a,b and h are not simultaneously zero is called the general equation of the second degree or the quadratic equation in x and y.

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 a,b,h.

Theorem: The general equation of the second degree a{x^2} = b{y^2} + 2hxy + 2gx + 2fy + c = 0 represents a conic section. It represents a:

(i) Parabola if {h^2} - ab = 0

(ii) Ellipse if {h^2} - ab < 0

(iii) Hyperbola if {h^2} - ab > 0

Theorem: If the axes are rotated about the origin through an angle \theta \left( {0 < \theta < {{90}^ \circ }} \right), where \theta is given by \tan 2\theta = \frac{{2h}}{{a - b}}, then the product terms xy in the general second degree equation vanish in the new coordinates' axes.

Example: Identify the conic represented by the equation 17{x^2} - 12xy + 8{y^2} = 0.

Comparing with the general equation of a conic we have a = 17, b = 8 and h = - 6, so

{h^2} - ab = {\left( { - 6} \right)^2} - \left( {17} \right)\left( 8 \right) = 36 - 136 = - 100 < 0

This shows that the given equation represents an ellipse.