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 determine which degenerate form or forms of the parabola cannot be obtained from the intersection of a plane and a double napped cone, you can use the diagram of a cone and the plane. Here's how to obtain each form:
1. Standard form: Start with a double napped cone, which is the typical conical shape. Then, intersect it with a plane that is parallel to one side of the cone. The resulting intersection will be a parabola in its standard form, y = ax^2 + bx + c.
2. Vertex form: Begin with a double napped cone. Next, intersect it with a plane that is perpendicular to the axis of the cone and passes through the tip or vertex of the cone. The resulting intersection will be a parabola in its vertex form, y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
3. Intercept form: Again, start with a double napped cone. Intersect it with a plane that is perpendicular to the axis of the cone, but not passing through the vertex. The resulting intersection will be a parabola in its intercept form, (x - h) = a(y - k)^2, where (h, k) represents the coordinates of the point through which the axis of the cone passes.
Now, to determine the degenerate form(s) that cannot be obtained, refer to the diagram:
- The degenerate form of a parabola that cannot be obtained is a straight line. This occurs when the plane intersects the cone parallel to the axis but intersecting both branches of the cone equally at the same height. In this case, the parabola degenerates into a line since all the points on the parabola have the same y-coordinate.
You can illustrate this with a diagram by drawing a double napped cone, then drawing a plane that intersects both branches of the cone parallel to the axis and at the same height. Label it as the degenerate form of a straight line.
Remember, the different forms of the parabola depend on the orientation and position of the plane's intersection with the double napped cone.