No, a conic section does not have vertices. If it is a circle, it has a center; if it is a parabola or hyperbola, it has a focus; and if it is an ellipse, it has foci.
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the figure defined by intersection of a cone and a plane.
the figure defined by intersection of a cone and a plane.
It sounds like this describes the conic section which is 2 straight lines intersecting at the origin [degenerate form of a hyperbola], but I may be misunderstanding the phrasing of the question.
It is a parabola.