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.