What degenerate form or forms of the parabola CANNOT be obtained from the intersection of a plane and a double napped cone? I have to use a diagram and give complete description how to obtain this ( or these) form(s)
but what diagram would i use???
To understand and visualize the different forms of a parabola that cannot be obtained from the intersection of a plane and a double napped cone, you can consider using a coordinate plane diagram. This will allow you to plot the different positions of the plane in relation to the cone.
Here's how you can create the diagram:
1. Draw a horizontal coordinate plane on a piece of paper or a digital drawing program.
2. On the plane, draw the outline of a double napped cone. You can draw a smooth, curved shape that opens upward and narrows towards the top.
3. Label the vertex of the cone as V and any other relevant points if needed.
4. Now, consider different positions of a plane in relation to the cone. Start with a plane that is parallel to the base of the cone and intersects it horizontally.
5. Draw this plane on the diagram. It should slice through the cone, creating a parabolic curve that opens upward.
6. Label this parabola as Form 1, which represents the standard form of a parabola obtained from the intersection of the plane with the cone.
7. Now, think about other orientations and positions of the plane.
a. For example, tilt the plane slightly so that it intersects the cone at an angle. Draw this plane on the diagram and observe the resulting parabolic curve.
b. If the plane intersects the cone at various angles and orientations, you will notice that the resulting parabolic curves are not of the standard form.
8. Identify and label these non-standard parabolas on the diagram. You can use different colors or patterns to distinguish them from the standard parabola.
9. The non-standard forms of the parabola are the degenerate forms that cannot be obtained from the intersection of a plane and a double napped cone.
By using this diagram, you can clearly show and explain the different forms of the parabola that cannot be obtained from the intersection of a plane and a double napped cone.