## 15.4 Outline

- Introduction
- definition
- model

- Geometric definition of conic sections
- parabola
- focus
- directrix
- axis
- vertex

- ellipse
- foci
- major axis
- center
- minor axis

- circle
- center
- radius

- hyperbola
- transverse axis
- center
- conjugate axis

- parabola
- Algebraic definition of conic sections
- general form
- first-degree equation
- second-degree equation

- line
- parabola
- ellipse
- circle
- hyperbola

- general form
- Graphing conic sections
- standard form
- ellipses
- standard-form equations
- equation of a circle
- horizontal ellipse
- vertical ellipse
- eccentricity
- aphelion
- perihelion

- applications

- hyperbolas
- standard-form equations
- vertices
- horizontal hyperbola
- vertical hyperbola
- length of axis
- applications

- Parabolic reflectors

## 15.4 Essential Ideas

Geometric definition of the conic sections:

A **parabola **is the set of all points in the plane equidistant from a given point (called the *focus*) and a given line (called the *directrix*).

An **ellipse **is the set of all points in a plane such that, for each point on the ellipse, the sum of its distances from two fixed points (called the *foci*) is a constant.

A **circle** (a special type of an ellipse) is the set of all points in a plane a given distance from a given point.

A **hyperbola** is the set of all points in a plane such that, for each point on the hyperbola, the difference of its distances from two fixed points (the *foci*) is a constant.

**Algebraic definition of the conic sections:**

The **general form** of the equation of a conic section is

*Ax*^{2}* + Bxy + Cy*^{2}* + Dx + Ey + F* = 0

where *A, B, C, D, E*, and *F* are real numbers and (*x, y*) is any point on the curve.

This equation is called a: **first-degree** equation if *A = B = C =*0

**second-degree** equation otherwise.

If *B = *0, then we classify the conic section as follows:

*A = C =*0,

then the conic section is a

**line**;

*A = *0 and *C* is not equal to 0 or if *A* is not equal to 0 and *C = *0, it is a **parabola**; *A *and *C* have the same sign, it is an **ellipse**;

*A = C* is a **circle**; and *A *and *C *have opposite signs, it is a **hyperbola.
**

If *B* is not equal to 0, then the conic is rotated.