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A plane intersects one nappe of a double-napped cone such that the plane is not perpendicular to the axis and is not parallel to the generating line. Which conic section is formed? O ellipse O hyperbola. O circle parabola

A plane intersects one nappe of a double-napped cone such that the plane is not perpendicular to the axis and is not parallel to the generating line. Which conic section is formed? O ellipse O hyperbola. O circle parabola

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Intersecting a Double-Napped Cone When a plane intersects one nappe of a double-napped cone such that the plane is not perpendicular to the axis and is not parallel to the generating line, it forms particular conic sections. **Question:** Which conic section is formed? **Answer Choices:** - Ellipse - Hyperbola - Circle - Parabola **Correct Answer:** Ellipse ### Explanation: A double-napped cone consists of two infinite cones that are mirror images of one another, sharing a common vertex. Conic sections are the curves obtained by intersecting a plane with this double-napped cone. 1. **Ellipse**: An ellipse is formed when the intersecting plane is angled such that it cuts through only one nappe of the cone and is not parallel to the base or the generating line and is not perpendicular to the axis. 2. **Hyperbola**: A hyperbola is formed when the plane intersects both nappes of the cone. 3. **Circle**: A circle is a specific case of an ellipse that forms when the intersecting plane is parallel to the base of the cone. 4. **Parabola**: A parabola is formed when the intersecting plane is parallel to the generating line of the cone. In this scenario, since the plane is neither perpendicular to the axis nor parallel to the generating line, the conic section formed is an **ellipse**. **Diagram:** There would ideally be a diagram here to illustrate this conic section, showing the cone, the intersecting plane, and the resultant ellipse.
Transcribed Image Text:### Intersecting a Double-Napped Cone When a plane intersects one nappe of a double-napped cone such that the plane is not perpendicular to the axis and is not parallel to the generating line, it forms particular conic sections. **Question:** Which conic section is formed? **Answer Choices:** - Ellipse - Hyperbola - Circle - Parabola **Correct Answer:** Ellipse ### Explanation: A double-napped cone consists of two infinite cones that are mirror images of one another, sharing a common vertex. Conic sections are the curves obtained by intersecting a plane with this double-napped cone. 1. **Ellipse**: An ellipse is formed when the intersecting plane is angled such that it cuts through only one nappe of the cone and is not parallel to the base or the generating line and is not perpendicular to the axis. 2. **Hyperbola**: A hyperbola is formed when the plane intersects both nappes of the cone. 3. **Circle**: A circle is a specific case of an ellipse that forms when the intersecting plane is parallel to the base of the cone. 4. **Parabola**: A parabola is formed when the intersecting plane is parallel to the generating line of the cone. In this scenario, since the plane is neither perpendicular to the axis nor parallel to the generating line, the conic section formed is an **ellipse**. **Diagram:** There would ideally be a diagram here to illustrate this conic section, showing the cone, the intersecting plane, and the resultant ellipse.
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