A parabola.
Then the cross-section is a circle or a point.
Hyperbola
A hyperbola.
A cylinder has a circular cross section that is parallel to its base.
If a right circular cone intersects a plane that runs parallel to the edge of the cone the result curve will be a parabola, unless the intersection includes the vertex of the cone, in which case the intersection is a straight line. This is a conic section. Depending on the angle of the plane, the section will be a circle, an ellipse, a parabola, or two hyperboles.
Then the cross-section is a circle or a point.
The "conic section" that is produced when a right circular cone intersects a plane that runs parallel to the edge of the cone is a parabola. In the case where the plane also intersects the vertex of the cone, the parabola becomes two intersecting lines.
Hyperbola
A hyperbola.
A cylinder has a circular cross section that is parallel to its base.
If a right circular cone intersects a plane that runs parallel to the edge of the cone the result curve will be a parabola, unless the intersection includes the vertex of the cone, in which case the intersection is a straight line. This is a conic section. Depending on the angle of the plane, the section will be a circle, an ellipse, a parabola, or two hyperboles.
If your question is "What is the cross-section of the intersection?" then the answer is "A circle." Otherwise, I can't make sense of the question.
The answer depends on the angle at which the axis of the cone intersects the cross-sections.
The question is incomplete, because "the following" was not provided. A circle, however, is a conic section where the sectioning plane is perpendicular to the cone's axis of symmetry and intersects each generator or, more specifically, if it is not a right circular cone, parallel to the generating circle of the cone.
It is a section of a right circular cone by a plane that is parallel to one generating line of the cone.
A cut through a right circular cylinder that is perpendicular to its altitude yields a circular cross-section. A right circular cylinder that is cut on a plane not perpendicular to its altitude but also but also not parallel to its altitude will yield an ellipse whose minor axis is the diameter of the cylinder. Trivial cases of a set of parallel lines, a single line, or the empty set occur when the cut is parallel to the altitude, externally tangent to the cylinder, or does not intersect the cylinder, respectively.
A circular section refers to a part or segment of a larger circular object or shape. It typically describes a section that is formed by cutting or dividing the circular object along a specific path or line. The circular section retains the characteristics and properties of the original circular shape.