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No; there is no physical way to intersect a plane and a double-napped cone to obtain parallel lines. The problem lies in getting a second line that doesn't intersect the first, since both cones intersect at a point, there is no way get a second line without intersecting the first line.

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A right circular double-napped cone is intersected by a plane that is parallel to the generator of the cone two conice sections that could be form are?

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