Identify the conic section defined as the set of all the points in the intersection of a plane and a cone when the plane is less steep than the generator line and is not perpendicular to the axis of symmetry of the cone.

A. Ellipse
Correct! When the slope of the plane is less steep than the slope of the generator line, it intersects one nappe of the cone to form an ellipse.

B. Hyperbola
Incorrect. A hyperbola is formed when the slope of the plane is steeper than the slope of the generator line.

C. Circle
Incorrect. A circle is formed when the plane is perpendicular to the vertical axis of the cone.

D. Parabola
Incorrect. A parabola is formed when the plane is parallel to the generator line. The plane has the same steepness as the generator line.

Identify the figure that best exemplifies the conic section which is the set of all the points in the intersection of a cone and a plane parallel to the generator line of the cone, intersecting one nappe.

A.

B.

C.

D.

A
Incorrect. Since the plane intersects both nappes of the cone, the plane is not parallel to the generator line. The conic described is not a hyperbola.

B
Correct! The plane intersects only one nappe of the cone and is parallel to the generator line. The conic described is a parabola.

C
Incorrect. Although the plane intersects only one nappe of the cone, it is not parallel to the generator line. The conic described is not an ellipse.

D
Incorrect. Although the plane intersects only one nappe of the cone, it is not parallel to the generator line. The conic described is not a circle.

One conic section is a special case of another. Select the option with the correct relationship.

A. A parabola is a special case of a hyperbola.
Incorrect. A hyperbola is formed when the slope of the plane is steeper than the slope of the generator line. The parabola is formed when the slope of the plane has the same slope as the generator line. Although a parabola may look like half of a hyperbola, they are two different types of figures.

B. A hyperbola is a special case of an ellipse.
Incorrect. A hyperbola is formed when the slope of the plane is steeper than the slope of the generator line. An ellipse is formed when the slope of the plane is less steep than the slope of the generator line. They are two different types of figures.

C. A circle is a special case of an ellipse.
Correct! All circles are special cases of ellipse. Both are formed when the slope of the plane that intersects the cone is less steep than the slope of the generator line.

D. A parabola is a special case of a circle.
Incorrect. A circle is formed when the slope of the plane is less steep than the slope of the generator line and is perpendicular to the axis of symmetry of the cone. The parabola is formed when the slope of the plane has the same steepness as the generator line. They are two different types of figures.

Select the definition that best describes an ellipse.

A. the conic section formed when a plane that slants with a slope less than the slope of the generator line intersects one nappe of a cone
Correct! An ellipse is described correctly by this definition.

B. the conic section formed when a plane that slants with a slope greater than the slope of the generator line intersects both nappes of a cone
Incorrect. This statement defines a hyperbola.

C. the conic section formed when a plane parallel (having the same slope) to the generator line intersects one nappe of a cone
Incorrect. This statement defines a parabola.

D. the conic section formed when a plane that is perpendicular to the axis of symmetry intersects one nappe of a cone
Incorrect. This statement defines a circle which is just one type specific type of ellipse.

Select the definition that best describes the conic section shown.

A. the conic section formed when a plane that slants with a slope less than the slope of the generator line intersects one nappe of a cone
Incorrect. This statement defines an ellipse and the figure shows a parabola.

B. the conic section formed when a plane that slants with a slope greater than the slope of the generator line intersects both nappes of a cone
Incorrect. This statement defines a hyperbola and the figure shows a parabola.

C. the conic section formed when a plane parallel (having the same slope) to the generator line intersects one nappe of a cone
Correct! This statement correctly defines a parabola like the one shown.

D. the conic section formed when a plane that is perpendicular to the axis of symmetry intersects one nappe of a cone
Incorrect. This statement defines a circle and the figure shows a parabola.

Which figure does not depict a degenerate conic section?

A.

B.

C.

D.

A
Incorrect . The plane intersects the cone so that its vertex is the only point of intersection. This figure shows a degenerate conic section.

B
Incorrect . The plane is tangent to the cone so that the line through the vertex is the only intersection. This figure shows a degenerate conic section.

C
Incorrect . The plane intersects both nappes of the cone so that it contains the vertex. This figure shows a degenerate conic section of two intersecting lines.

D
Correct! The plane intersects both nappes of the cone but does not contain the vertex. Although the grid appears to picture two intersecting lines, the graph shows a hyperbola with the vertices close to each other.