Trying to understand what the conic section elements are

Question

If possible, could someone explain the major elements of the conic sections? By that I mean

• focus
• directrix
• principal axis
• vertex
• eccentricity
• and any other features of a conic section that are important.

My precalculus textbook is unclear.

[And here is where I (OP) should put what I do know]
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Answer 1

Ellipse

Take two points $F_1$ and $F_2$ in the plane and fix a number $L$ that's bigger than the distance between the two points. Then an ellipse is is the set of points $P$ such that the distance from $F_1$ to $P$ plus the distance from $F_2$ to $P$ is $L$.

$F_1$ and $F_2$ are called the foci of the ellipse. Besides $L$, we define a few other useful numbers associated with ellipses.

Notice that in general ellipses has a longer direction (horizontal in the image above) and a shorter direction (vertical).

• The line segment through the center of the ellipse to each of the furthest points is called the major axis. Half of this (the line segment from the center to just one of the furthest points) is a semi-major axis.
• The other axis is the minor axis and it has two semi-minor axes, as well.
• Those two further points on the ends of the major axis are called the vertices (plural of vertex) and the end points of the minor axis are called the co-vertices (plural of co-vertex).
• The two points we used to draw our ellipse are called the foci (plural of focus).
• The focal distance is the distance from the center to one of the foci.
• The eccentricity describes how elongated the ellipse is. It is calculated as the focal length divided by the length of the semi-major axis. So it is the fraction of the length of the semi-major axis that the focal length takes up (think percentage but in decimal form).
  • The smallest the eccentricity can be is $0$.The is when the foci are right at the center. In this case you just get a circle.
  • Going the other way, as the focal length gets closer and closer to the length of the semi-major axis -- i.e., the eccentricity gets closer to $1$ -- the less circular and more elongated the ellipse looks.

Parabola

Estimated release date: 12/15/2017.

Hyperbola

Plans are to release it as DLC sometime next year.