The point at which a tangent meets a curve is called the point of contact between the tangent and the curve. The following figures show the point of contact of a tangent and a parabola, circle, ellipse and hyperbola respectively.

Point of Contact of an Ellipse and its Tangent 

Point of Contact of a Parabola and its Tangent 

Point of Contact of a Circle and its Tangent 
A tangent may have more than one point of contacts to a curve, of which the best example is the sine curve having a horizontal tangent at its upper or lower bound (shown in the diagram below).

Points of contact of sine curve with a horizontal tangent 
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