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Q: What is the intersection of a plane passing through the apex of a right circular cone?

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Then the intersection is a hyperbola.

The intersection forms a hyperbola.

rectangle

A parabola is the figure formed by the intersection of a circular cone and a plane that lies parallel to the edge of the cone. (the cone does not have to be a right [90°] circular cone).

Section plane is the intersection of a plane cutting through a solid

Related questions

Then the intersection is a hyperbola.

The intersection forms a hyperbola.

rectangle

A parabola is the figure formed by the intersection of a circular cone and a plane that lies parallel to the edge of the cone. (the cone does not have to be a right [90°] circular cone).

The intersection of a right circular cone and a plane that is parallel to the edge of the cone is a parabola. However, if the vertex of the cone lies on the plane, then the intersection is simply two intersecting lines.

If the line is not IN the plane ... it just zaps through the plane from some direction ... then it touches the plane in only one point. The intersection is a point.if it is lined up with the plane, then the intersection is a line.

If a right circular cone is intersected by a plane so that the intersection goes through the cone's vertex as well as an edge of each nappe, the shape produced is a line. Not asked, but... If the angle of the plane is less than the angle of the cone, then the intersection is a point. If the angle of the plane is greater than the angle of the cone, then the intersection is two lines intersecting at the vertex. If the plane insersects at other than the vertex, then the intersection is a circle when the plane is perpendicular to the cone's axis, an ellipse when the plane's angle is less than the cone's angle, a parabola when the planes's angle equals the cone's angle, and two hyperbole's in the last case.

If I understand your description correctly, a line.

not quite a cirlcel but an oblong circle.

Section plane is the intersection of a plane cutting through a solid

The intersection of the cone and that particular plane is a parabola.

The intersection will consist of only one point.