Discriminant Of a Conic Section

Any second-degree curve equation can be written as $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ or $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ where $$A,B,C,D,E,E,a,b,c,f,g,h\in\mathbb R$$

To find type of conic and nature of conic we use $\Delta$ and is given by $$\Delta=\begin{vmatrix}a&h&g\\ h&b&f\\ g&f&c\end{vmatrix}$$ $$=abc+2fgh-af^2-bg^2-ch^2$$

If $\Delta$ is $0$, it represents a degenerate conic section, otherwise, it represents a non-degenerate conic section.

Also, the type of conic section that the above equation represents can be found using the discriminant of the equation, which is given by $B^2-4AC$.

Conditions regarding the quadratic discriminant are as follows:

If $\Delta=0$:

$\bullet$ If $h^2-ab\gt0$, the equation represents two distinct real lines.

$\bullet$ If $h^2-ab=0$, the equation represents parallel lines.

$\bullet$ If $h^2-ab\lt0$, the equation represents non-real lines.

If $\Delta\neq0$:

$\bullet$ If $B^2-4AC\gt0$, it represents a hyperbola and a rectangular hyperbola $(A+C=0)$.

$\bullet$ If $B^2-4AC=0$, the equation represents a parabola.

$\bullet$ If $B^2-4AC\lt0$, the equation represents a circle $(A=C,B=0$) or an ellipse $(A\neq C)$. For a real ellipse, $\Big(\frac\Delta{a+b}\lt0\Big)$.