Conic Sections

Conic section is the intersection between a cone and a plane. To make the explanation simple, I use an upright cone for the following examples. There are basically three types of conic sections:

1. Ellipse
2. Parabola
3. Hyperbola

Let's take a look at how the conic sections are formed:

1. Ellipse
   An ellipse is formed when a slanted plane ax + by + cz = d intersects with the upright cone. Take note the angle of the plane from the z-axis is bigger than the angle of the cone from the z-axis. Otherwise, a hyperbola is form.
   When a, b = 0, the plane becomes a horizontal plane, cz = d. This makes the curve of intersection changes from the ellipse to a circle.
2. Parabola
   A parabola is formed when a plane that is "parallel" to the cone intersects with the cone as shown in the first graph below.
3. Hyperbola
   When a plane ax + by + cz = d (slanted/vertical) that forms an angle from the z-axis is smaller than the angle of the cone from the z-axis (as mentioned in ellipse just now), a hyperbola is formed.
   

Betty, Created with GeoGebra