Instructions:Answer all the following questions in the space provided. Simplify all answers.
- Describe or show how a double-napped cone is created.
A generator is rotated about a fixed vertical axis.
- Label the vertex, the vertical axis, and the generator in the following diagram of a double-napped cone.
- a) Describe or show how the double-napped cone can be sliced to create circles.
To create circles, the double-napped cone must be sliced by a plane that is perpendicular to the vertical axis of the cone.
b) Describe what happens to a circle as the plane cutting the double-napped cone moves:
i) towards the cone's vertex
the radius of the circle decreases; the circle gets smaller
ii) away from the cone's vertex
the radius of the circle increases; the circle gets bigger
- a) Describe or show how the double-napped cone can be sliced to create ellipses.
To create ellipses, the double-napped cone must be sliced by a plane that only intersects one nap of the cone. That is, the angle of the plane with the vertical axis must be greater than the angle between the generator and the vertical axis, and less than 180o minus this angle. Also, the plane must not intersect the cone at its vertex.
b) Describe what happens to an ellipse as the plane cutting the double-napped cone gradually becomes:
i) more parallel to the cone's generator.
the ellipse becomes more elongated; the lengths of the horizontal and vertical axes become more different
ii) more perpendicular to the cone's generator.
the ellipse becomes more circular; the lengths of the horizontal and vertical axes become more similar
- a) Describe or show how the double-napped cone can be sliced to create hyperbolas.
To create hyperbolas, the double-napped cone must be sliced by a plane that intersects both naps of the cone. That is, the angle of the plane with the vertical axis must be less than the angle between the generator and the vertical axis, or greater than 180o minus the angle between the generator and the vertical axis.
b) Describe what happens to a hyperbola as the plane cutting the double-napped cone moves:
i) closer to the cone's vertex.
the vertices of the hyperbola become closer together
ii) farther away from the cone's vertex.
the vertices of the hyperbola grow farther apart
- a) Describe or show how the double-napped cone can be sliced to create parabolas.
To creat parabolas, the double-napped cone must be sliced by a plane that is exactly parallel to the cone's generator.
b) Describe what happens to a parabola as the plane cutting the double-napped cone moves:
i) closer to the cone's vertex.
the parabola becomes narrower
ii) farther away from the cone's vertex.
the parabola becomes wider
- a) Use the below chart to describe or show how the double-napped cone can be sliced to create each of the degenerate cases.
| Degenerate Conic |
Describe or show how this Degenerate Case is Created |
|---|
| a point |
the cone is cut by any plane going through its vertex but not containing the generator or the vertical axis
|
| a line |
the cone is cut by a plane containing the generator
|
| intersecting lines |
the cone is cut by a plane containing the cone's vertical axis
|
- State the general equation that describes all the conic sections and degenerate conics algebraically.
Ax2 + By2 + Cx + Dy + F = 0
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