The answer will depend on the angle formed between the plane and the axis of the cone. Since there is no picture "below" it is not possible to determine that and therefore it is impossible to give an answer.
A point
An Ellipse
A point.
It will be a hyperbola.
hyperbola
A spitial figure with a circular bases no edges and one vertex
A line is produced
The intersection will consist of only one point.
An Ellipse
A point.
If I understand your description correctly, a line.
It will be a hyperbola.
hyperbola
hyperbola
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.
Acute angles.
A spitial figure with a circular bases no edges and one vertex
a cone