# 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 amYou 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 pmAs 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.