For elliptic curves, how to express the condition at which the vertical tangent line L, to curve E, at point P is at infinity point?

Question

So I'm writing an essay for my school program. I'm trying to convey the two cases at which P + Q = (a point at infinity). I've written the following for case 1:

(1) P != Q and x_p = x_q then P + Q = (point at infinity)

(2) This is the case that I don't know how to convey best. I've seen one maths paper say that in this case when P is a vertical tangent to the elliptic curve E and doesn't intersect any other points, then you can say P = Q and x_p = 0 then P + Q or P + P = (point at infinity). But what confuses me is that, doesn't the condition of x_p, just cover the case in which the vertical tangent to E is at x-coordinate zero, but doesn't necessarily cover for when the vertical tangent to E is at other x coordinates, or am I mistaken?

(3)In another paper I read that they said y_p = y_p = 0 then P + P = (point at infinity) and that left me even more confused. Because that one I don't agree with at all.