Essential Concepts
- The equation of a vertical parabola in standard form with given focus and directrix is [latex]y=\frac{1}{4p}{\left(x-h\right)}^{2}+k[/latex] where p is the distance from the vertex to the focus and [latex]\left(h,k\right)[/latex] are the coordinates of the vertex.
- The equation of a horizontal ellipse in standard form is [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex] where the center has coordinates [latex]\left(h,k\right)[/latex], the major axis has length 2a, the minor axis has length 2b, and the coordinates of the foci are [latex]\left(h\pm c,k\right)[/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex].
- The equation of a horizontal hyperbola in standard form is [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex] where the center has coordinates [latex]\left(h,k\right)[/latex], the vertices are located at [latex]\left(h\pm a,k\right)[/latex], and the coordinates of the foci are [latex]\left(h\pm c,k\right)[/latex], where [latex]{c}^{2}={a}^{2}+{b}^{2}[/latex].
- 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 [latex]r=\frac{ep}{1\pm e\cos\theta }[/latex] or [latex]r=\frac{ep}{1\pm e\sin\theta }[/latex], where p represents the focal parameter.
- To identify a conic generated by the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[/latex], first calculate the discriminant [latex]D=4AC-{B}^{2}[/latex]. If [latex]D>0[/latex] then the conic is an ellipse, if [latex]D=0[/latex] then the conic is a parabola, and if [latex]D<0[/latex] 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 [latex]4AC-{B}^{2}[/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[/latex], 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
