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Which of the following conic sections could be formed when a plane intersects both cones of a double cone? 2 0- -6 -6 -5 -3 -1 12 -10 -1 -3 -6

Which of the following conic sections could be formed when a plane intersects both cones of a double cone? 2 0- -6 -6 -5 -3 -1 12 -10 -1 -3 -6

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter13: Conic Sections
Section13.4: Hyperbolas
Problem 55PS
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--- ### Analyzing Conic Sections from Plane Intersections with a Double Cone **Question:** Which of the following conic sections could be formed when a plane intersects both cones of a double cone? **Graphs and Explanations:** 1. **Upper Graph:** - **Description:** The upper graph displays a circle centered at the origin of the coordinate plane. - **Graph Analysis:** The circle has a radius of 5 units as it extends from -5 to 5 on both the x-axis and y-axis. The circle is a closed curve where every point is equidistant from the center. 2. **Lower Graph:** - **Description:** The lower graph illustrates a parabola that opens upwards. - **Graph Analysis:** The vertex of the parabola is at the origin (0, 0). The parabola is symmetric about the y-axis and extends infinitely in the positive y direction. The arms of the parabola widen as they move away from the vertex. **Interpreting the Conic Sections:** - A **circle** is formed when a plane intersects a double cone parallel to the base of the cone. - A **parabola** is formed when a plane intersects the double cone parallel to the slant height of one of the cones and passes through only one of the cones. **Conclusion:** Both graphs represent conic sections that can be formed when a plane intersects a double cone. The circle is formed under specific conditions different from those needed to form a parabola. Understanding these intersection properties helps illustrate the relation and parameters involved in conic sections. --- This descriptive analysis of the graphs aims to aid in the visual understanding of conic sections and their formation through geometric intersections.
Transcribed Image Text:--- ### Analyzing Conic Sections from Plane Intersections with a Double Cone **Question:** Which of the following conic sections could be formed when a plane intersects both cones of a double cone? **Graphs and Explanations:** 1. **Upper Graph:** - **Description:** The upper graph displays a circle centered at the origin of the coordinate plane. - **Graph Analysis:** The circle has a radius of 5 units as it extends from -5 to 5 on both the x-axis and y-axis. The circle is a closed curve where every point is equidistant from the center. 2. **Lower Graph:** - **Description:** The lower graph illustrates a parabola that opens upwards. - **Graph Analysis:** The vertex of the parabola is at the origin (0, 0). The parabola is symmetric about the y-axis and extends infinitely in the positive y direction. The arms of the parabola widen as they move away from the vertex. **Interpreting the Conic Sections:** - A **circle** is formed when a plane intersects a double cone parallel to the base of the cone. - A **parabola** is formed when a plane intersects the double cone parallel to the slant height of one of the cones and passes through only one of the cones. **Conclusion:** Both graphs represent conic sections that can be formed when a plane intersects a double cone. The circle is formed under specific conditions different from those needed to form a parabola. Understanding these intersection properties helps illustrate the relation and parameters involved in conic sections. --- This descriptive analysis of the graphs aims to aid in the visual understanding of conic sections and their formation through geometric intersections.
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