Answer:
1) Circle 2) Parabola 3) A plane intersects only one nappe of a cone, and a plane is parallel to the generating line of the cone. 4) A degenerate Hyperbola is a pair of intersecting lines.
Step-by-step explanation:
1) A plane intersects one nappe of a double-napped cone such that it is perpendicular to the vertical axis of the cone. Which conic section is formed?
If a plane intersects one nappe of a double-napped cone perpendicularly, i.e. 90º then clearly the conic section is going to produce a circle.
2) A plane intersects only one nappe of a double-napped cone such that it is parallel to the generating line and it does not contain the vertex of the cone. Which conic section is formed?
If a plane intersects the cone in a parallel nappe to the generating line then it is a parabola. Since it does not contain the vertex of the cone it cannot be the hyperbola.
3) Which intersection forms a parabola?
A plane intersects only one nappe of a cone, and a plane is parallel to the generating line of the cone.
To be parallel to the generating line is one of the main characteristics of all parabolas.
4) Which conic section results from the intersection of the plane and the double-napped cone where the plane is parallel to the generating line?
It's a parabola due to be parallel to the generating line.
Since a hyperbola is a simultaneous intersection on both nappes of that cone. Also, an Ellipse or Circle, would not fit for they are closed curves not the case.
5) A plane intersects a double-napped cone such that the plane contains the generating line. Which terms describe the degenerate conic section that is formed?
A degenerate Hyperbola is a pair of intersecting lines.
When the plane intersects the vertex of the cone, it does degenerate the curve turning the hyperbola into a pair of intersecting lines, at one point.
