General form

Definition

General form of a conic section is given by the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. It represents parabolas, ellipses, and hyperbolas depending on the values of coefficients.

5 Must Know Facts For Your Next Test

1. The general form can represent all types of conic sections: parabolas, ellipses, and hyperbolas.
2. If $B^2 - 4AC < 0$, the conic is an ellipse; if $B^2 - 4AC = 0$, it is a parabola; if $B^2 - 4AC > 0$, it is a hyperbola.
3. In this form, $A$, $B$, and $C$ determine the shape and orientation of the conic section.
4. The general form can be converted to standard form to identify specific properties like foci, vertices, and axes.
5. The discriminant ($B^2 - 4AC$) is key in determining the type of conic section represented.

Review Questions

• What does the discriminant ($B^2 - 4AC$) tell you about a conic section?
• How can you convert a general form equation into its standard form?
• What type of conic section is represented if $A = C$ and $B = 0$?

Related terms

Discriminant:

In the context of conic sections, it is given by $B^2 - 4AC$ and determines whether the conic is an ellipse, parabola, or hyperbola.

Standard Form: The simplified version of a conic section's equation that makes it easier to identify its properties.

Conic Sections:

Curves obtained by intersecting a plane with a double-napped cone; includes ellipses, parabolas, and hyperbolas.