One of the things I inherited over my decades of teaching mathematics
is a wooden right circular cone.
You can likely tell from the photo (left, below) that it has been wellused and muchloved.
Of course, cones can be oriented in many different ways!
To simplify language, in this section we assume cones are oriented as shown below—like an upsidedown icecream cone
sitting on a table.
In this position, a horizontal slice chops off
the top of the cone, leaving you looking down at a small circle.
(This sketch is from Wikipedia.) 
Look at the coloradorned image in the middle, and you'll see the various cuts where parts of my wooden cone
can be removed.
As we'll discuss on this page, these ‘slices’
illustrate the conic sections:
When I present this wooden cone to my students, they need a good imagination.
Why? Because what I really want them to envision is this:
In the sections on conics, we'll call this an infinite double cone. Think of it as a sort of infinite hourglass. There are different ways to define conic sections, but the definition we're using here is this:
Conic sections are curves that are
formed by the intersection of a plane with the surface of an infinite double cone.


Most people like to think of the conic sections as:
However, if the slicing plane goes through the apex, then weird things can happen!
These resulting intersections are called degenerate conics.
For example, if a horizontal plane passes through the apex, then the intersection is a single point.
How boring! People don't typically want to call a point a conic section!
So, a point is an example of a degenerate conic.
Here's a fantastic youtube video on conic
sections.
Its length is 5:27 (five minutes, 27 seconds): if you have limited time, skip the intro/terminology and start at 1:38.
Degenerate conics start at 3:47, and you'll see that there
are three types:
So, why is the name ‘conic’ or ‘conic section’ appropriate?
Because conics arise by intersecting a plane with an (infinite double) CONE!!
Circle
Start with a horizontal plane cutting through the double cone
(anywhere except through the apex). 

Ellipse
Now, start tipping the plane slightly from the horizontal. 

Parabola
As you continue to tilt the plane, you'll reach an instant
when the plane is parallel to the ‘edge/side’ of the cone. 


Hyperbola
Continue to tilt the plane, so it's getting steeper and steeper (approaching vertical). 

There are many occurrences of conic shapes in nature and life.
Here are some examples:
Also, conics have special properties that make them valuable tools.
Here are some examples:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
IN PROGRESS 