#### Common Parts of Conic Sections

A **focus **is a point about which the conic section is constructed. In other words, it is a point about which rays reflected from the curve converge. A parabola has one focus about which the shape is constructed; an ellipse and hyperbola have two.

A **directrix **is a line used to construct and define a conic section. The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.

#### Eccentricity

- 0 <
**eccentricity**< 1 we get an ellipse, **eccentricity**= 1 a parabola, and**eccentricity**> 1 a hyperbola.

A circle has an **eccentricity of zero**, so the eccentricity shows us how “un-circular” the curve is. The bigger the eccentricity, the less curved it is.

#### Porabolla

A parabola is a curve where any point is at an **equal distance** from:

- a fixed point (the
**focus**), and - a fixed straight line (the
**directrix**)

#### Ellipse

“F” is a **focus**, “G” is a **focus**,and together they are called **foci** (pronounced “fo-sigh”).

The total distance from **F to P to G** stays the same

Well f+g is equal to the **length of the major axis**.

#### Hyperbola

A hyperbola is two curves that are like infinite bows.

Looking at just one of the curves:

any point **P** is closer to **F** than to **G** by some constant amount

The other curve is a mirror image, and is closer to G than to F.

- an
**axis of symmetry**(that goes through each focus) - two
**vertices**(where each curve makes its sharpest turn) - the distance between the vertices (2a on the diagram) is the
**constant difference**between the lengths**PF**and**PG** - two
**asymptotes**which are not part of the hyperbola but show where the curve would go if continued indefinitely in each of the four directions

#### Hyperbolic functions

**Catenary (hanging cable)**