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.
Galada
August 2 @ 1:52 am
You do not mention the two line solution to the general 2nd degree equation. A textbook says that if, when solving for x, for a two line solution the quantity under the square root sign must be a perfect square. I do not see why this has to be so. galada
OldProf
March 21 @ 8:08 pm
As Galada has pointed out, this page omitted an entire class of conic section: a pair of straight lines. The page, despite being sketchy, started out (and continued) confusingly with a wrong equation. The first “=” between the x^2 and y^2 terms should be a “+”. To answer Galada’s question, for the equation to represent a pair of straight lines, the left-hand-side of the conic eq (with the “=” fixed) must be reducable into 2 linear factors. The “quantity under the sq root” Galada mentioned refers to the discriminant of the eq when rearranged as a quadratic equation in terms of x or y. When the conic eq is viewed as a quadratic equation in terms of say, y, the discriminant would be an expression in terms of x. For the conics eq to represent a pair of straight lines, the ‘roots’ of the quadratic equation have to be in the form: y=Ax+B. Hence the discriminant of that quadratic must be a perfect square (for that discriminant expression in x), so that the sq-root could be reduce into a linear expression.
American Citizen and math tutor
March 18 @ 11:56 am
More supposedly true math information posted to the internet, but riddled with errors. I have spent the last 3 days on the internet trying to find out how to correctly go from the general form (given incorrectly above) to the standard one for an ellipse, and all I find (even from universities which should know better) is half-solutions, nearly everyone omitting the xy term and yet all acting as if they are the last word. It truly is almost appalling to see this true lack of true knowledge being display worldwide to this degree. We seem to have been miseducated into confusion.