What is formed at the intersection of two curves or a curve and a line of opposite direction?

The correct response recognizes that a point of cusp is where two curves meet in such a way that they share a common point but have different directions at that point. This intersection usually happens when the two curves approach each other but do not smoothly join, creating a sharp point. Typically, this results in a point that is often associated with a change in the curve's direction, leading to a cusp.

In contrast, the other terms represent different concepts in geometry. A point of tangency occurs when a curve touches a line or another curve but does not cross it, and both share the same slope at that point, resulting in smooth continuity rather than a sharp change. A vertex is typically used to denote a point where two lines meet in a polygon, like at the corners of a triangle. The focus refers to a specific point in conic sections that has unique properties related to the curve's shape, like that of a parabola or ellipse. Thus, the uniqueness of the cusp lies in its representation of an abrupt directional change at a point where two curves intersect.