Which term describes the red curve in the figure below? A. Hyperbola B. Parabola C. Circle D. Ellipse ? PREVIOUS
The Correct Answer and Explanation is:
The correct answer is:
D. Ellipse
Explanation:
The figure shows a double-napped cone (two cones stacked tip-to-tip), intersected by a green plane that cuts through only one nappe (one cone) at an angle oblique to the axis and not parallel to the side of the cone. The result of this intersection is a red curve, which is an ellipse.
This scenario belongs to the study of conic sections, which are the curves obtained by slicing a cone with a plane in different ways. The type of curve formed depends on the angle and position of the intersecting plane:
- Circle: A special case of an ellipse, formed when the plane cuts perpendicular to the axis of the cone (i.e., horizontally across one nappe). This is not the case in the figure because the plane is tilted.
- Ellipse: Formed when the plane cuts obliquely through one nappe and does not intersect the base, and is not parallel to any edge of the cone. This matches the given figure perfectly, as the plane cuts across one cone diagonally but does not go through both cones or align with their sides.
- Parabola: Formed when the plane is parallel to the slant of the cone. The red curve would then be infinitely long on one side, not a closed shape. This is clearly not the shape shown.
- Hyperbola: Formed when the plane cuts through both nappes of the double cone. This results in two separate open curves. Again, the image shows only one closed curve on a single nappe.
Therefore, based on the orientation of the green plane and the shape of the resulting red curve (a closed, elongated loop), the correct classification is:
D. Ellipse.
This highlights the fundamental geometric property that ellipses are closed conic sections formed by slicing through one nappe of a cone at an angle.
