Essential Concepts
- The equation of a vertical parabola in standard form with given focus and directrix is where p is the distance from the vertex to the focus and are the coordinates of the vertex.
- The equation of a horizontal ellipse in standard form is where the center has coordinates , the major axis has length 2a, the minor axis has length 2b, and the coordinates of the foci are , where .
- The equation of a horizontal hyperbola in standard form is where the center has coordinates , the vertices are located at , and the coordinates of the foci are , where .
- The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.
- The polar equation of a conic section with eccentricity e is or , where p represents the focal parameter.
- To identify a conic generated by the equation , first calculate the discriminant . If then the conic is an ellipse, if then the conic is a parabola, and if then the conic is a hyperbola.
Glossary
- conic section
- a conic section is any curve formed by the intersection of a plane with a cone of two nappes
- directrix
- a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two
- discriminant
- the value , which is used to identify a conic when the equation contains a term involving , is called a discriminant
- eccentricity
- the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix
- focal parameter
- the focal parameter is the distance from a focus of a conic section to the nearest directrix
- focus
- a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two
- general form
- an equation of a conic section written as a general second-degree equation
- major axis
- the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis
- minor axis
- the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis
- nappe
- a nappe is one half of a double cone
- standard form
- an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes
- vertex
- a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch
Candela Citations
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- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction