12.1 The Ellipse

An ellipse is the set of all points $(x, y)$ in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See Example 1 and Example 2.
When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See Example 3 and Example 4.
When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See Example 5 and Example 6.
Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See Example 7.

12.2 The Hyperbola

A hyperbola is the set of all points $(x, y)$ in a plane such that the difference of the distances between $(x, y)$ and the foci is a positive constant.
The standard form of a hyperbola can be used to locate its vertices and foci. See Example 1.
When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See Example 2 and Example 3.
When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See Example 4 and Example 5.
Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See Example 6.

A parabola is the set of all points $(x, y)$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
The standard form of a parabola with vertex $(0, 0)$ and the x-axis as its axis of symmetry can be used to graph the parabola. If $p > 0,$ the parabola opens right. If $p < 0,$ the parabola opens left. See Example 1.
The standard form of a parabola with vertex $(0, 0)$ and the y-axis as its axis of symmetry can be used to graph the parabola. If $p > 0,$ the parabola opens up. If $p < 0,$ the parabola opens down. See Example 2.
When given the focus and directrix of a parabola, we can write its equation in standard form. See Example 3.
The standard form of a parabola with vertex $(h, k)$ and axis of symmetry parallel to the x-axis can be used to graph the parabola. If $p > 0,$ the parabola opens right. If $p < 0,$ the parabola opens left. See Example 4.
The standard form of a parabola with vertex $(h, k)$ and axis of symmetry parallel to the y-axis can be used to graph the parabola. If $p > 0,$ the parabola opens up. If $p < 0,$ the parabola opens down. See Example 5.
Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See Example 6.

Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
A nondegenerate conic section has the general form $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ where $A, B$ and $C$ are not all zero. The values of $A, B,$ and $C$ determine the type of conic. See Example 1.
Equations of conic sections with an $x y$ term have been rotated about the origin. See Example 2.
The general form can be transformed into an equation in the $x^{'}$ and $y^{'}$ coordinate system without the $x^{'} y^{'}$ term. See Example 3 and Example 4.
An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See Example 5.

Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus $P (r, θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
A conic is the set of all points $e = \frac{P F}{P D},$ where eccentricity $e$ is a positive real number. Each conic may be written in terms of its polar equation. See Example 1.
The polar equations of conics can be graphed. See Example 2, Example 3, and Example 4.
Conics can be defined in terms of a focus, a directrix, and eccentricity. See Example 5 and Example 6.
We can use the identities $r = \sqrt{x^{2} + y^{2}}, x = r \cos θ,$ and $y = r \sin θ$ to convert the equation for a conic from polar to rectangular form. See Example 7.