## Summary of Conic Sections

### Essential Concepts

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