Conic sections

Definition

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These curves include circles, ellipses, parabolas, and hyperbolas.

5 Must Know Facts For Your Next Test

1. The general equation for conic sections in Cartesian coordinates is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
2. Conic sections can be derived using parametric equations, which express the coordinates of the points on the curve as functions of a parameter.
3. Each type of conic section has unique properties: circles have constant radius, ellipses have two foci, parabolas have a directrix and focus, and hyperbolas have asymptotes.
4. In trigonometry, parametric equations for conics often involve sine and cosine functions to represent circular or elliptical orbits.
5. Transformations such as rotation and translation can convert the general form of a conic section into its standard form.

Review Questions

• What is the general equation for conic sections in Cartesian coordinates?
• How do parametric equations relate to conic sections?
• What are the distinguishing features of each type of conic section?

Related terms

Ellipse: A type of conic section that forms an oval shape and has two foci such that the sum of distances from any point on the ellipse to the foci is constant.

Parametric Equations: Equations that express the coordinates of points on a curve as functions of one or more parameters rather than as explicit relationships between x and y.

Focus (Foci): Special points used in defining and constructing various types of conic sections; for example, an ellipse has two foci while a parabola has one.