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The intersection of two or more mathematical objects is the set of all points that are common to all of them. In set theory, that would be the elements in common. In geometry, it would be the set of all points in common. For example, the intersection of two different planes is a line; the intersection of a plane and a cone are the conic sections: circle, ellipse, parabola and hyperbola.
the other one is intersection
It is their intersection.
The intersection of two planes is never a point. It's usually a line. But if the planes have identical characteristics, then their intersection is a plane. And if the planes are parallel, then there's no intersection.
Not possible, the point would be part of the line. * * * * * So, the intersection is that point. Think in terms of sets rather than geometry.