### Key Concepts

## 12.1 The Ellipse

- An ellipse is the set of all points $\left(x,y\right)$ 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 $\left(x,y\right)$ in a plane such that the difference of the distances between $\left(x,y\right)$ 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.

## 12.3 The Parabola

- A parabola is the set of all points $\left(x,y\right)$ 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 $\left(0,0\right)$ 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 $\left(0,0\right)$ 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 $\left(h,k\right)$ 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 $\left(h,k\right)$ 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.

## 12.4 Rotation of Axes

- 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 $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ 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 $xy$ term have been rotated about the origin. See Example 2.
- The general form can be transformed into an equation in the ${x}^{\prime}$ and ${y}^{\prime}$ coordinate system without the ${x}^{\prime}{y}^{\prime}$ 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.

## 12.5 Conic Sections in Polar Coordinates

- 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 $P(r,\theta )$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
- A conic is the set of all points $e=\frac{PF}{PD},$ 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 $r=\sqrt{{x}^{2}+{y}^{2}},x=r\phantom{\rule{0.8em}{0ex}}\text{}\mathrm{cos}\phantom{\rule{0.4em}{0ex}}\text{}\theta ,$ and $y=r\phantom{\rule{0.8em}{0ex}}\text{}\mathrm{sin}\phantom{\rule{0.4em}{0ex}}\text{}\theta $ to convert the equation for a conic from polar to rectangular form. See Example 7.