My teacher told me that we are mistaken coming out of school that tangent tocuhes a point at one point. According to him, a tangent is just a special type of secant where two points share the same position but actually are just two different points.
My friend said it could happen if we think of tangent as a secant where the two points tend to each other.
I am dissatisfied with it. According to my teacher, the points share the location but are different. Since a point has only its position as its property, how are they differentiated among themselves? E.g. if we take the set of points on which tangent touches the curve (in a sufficiently small neighbourhood around the points), we get a single point since a set doesn't allow redundant entries.
(Moreover, my teacher's tried to explanation his position using quadratic formula taking a quadratic equation as example; but I fail to recognise its relevance except that a quadratic equation has at most two roots, and maybe that tangent has a solution with the curve. But isn't it the problem to begin with, how many solutions do the two of them have?)
And I could agree with my friend too, but if we take a curve to be $y = x^2$ and tangent to be $y = 0$ than there is no point except $(0,0)$ that is common to them. No point in any small neighbourhood around $(0,0)$ is common to both. That seems to only reinforce myself.
How many points do a tangent and its curve actually share? And if they are more than one, how can they be differentiated? That is, what is the application of treating two positions with same position differently?