What's the relationship between each conic section ( Parabola, circle, hyperbola, and ellipse) and a cone?

If you slice a cone with a plane, the intersection of the plane and cone will generate one of the conic sections (hence the name). Imagine a cone with it's point on the ground and its axis pointing vertically up. Let's call the angle of the cone surface relative to the ground as angle alpha. If you slice the cone at an angle parallel to the ground somewhere above the ground you get a circle. If you slice at an angle between parallel to the ground and alpha you get an ellipse. If you slice precisely at angle alpha you get a parabola. And if you slice at an angle that is greater than alpha and less than or equal to the vertical you get a hyperbola. Thus each of the conic shapes can be generated by slicing the code at different angles.