When it intersect with only 2 points, one of those points is intersected "twice": the line is tangent to the curve. You could also see the tangent as the limit

When it intersect with only 2 points, one of those points is intersected "twice": the line is tangent to the curve. You could also see the tangent as the limit when you intersect with two points, and then draw one of those points towards the other, closer and closer. In practice, this means that P+P doesn't compute exactly the same way as P+Q where P and Q are different points. But in practice it really does mean the same thing.