Elliptic Curve Non-tangent Non-Vertical Line

Question

If I plot a slightly non-vertical line that's not a tangent, it seems that it would intersect an elliptic curve at only 2 points? Is this correct?

If so it throws me off my understanding the explanations that say that you always intersect 3 points (with multiplicity at tangents).

With an exactly vertical line, it seems that the second point is an inverse while the third point is the O; i. e. P + -P + O = O

However if the line is slightly non-vertical it gives P + -Q + O = O which seems fishy because P has multiple inverses? What am I understanding wrong?

Update: As correctly pointed out in the answer and comment the third point does exist but it just plots way outside the plot area. I managed to plot it on Wolfram Alpha and the two attached images show what happens with a quite diagonal line and with a quite vertical line, for the same elliptic curve. Here is the image of a diagonal line

and here is the almost vertical one.

Also, another way I convinced myself is that for any non-vertical line, y grows linearly with x whereas the elliptical curve grows super-linearly with x, and hence faster than the line. So eventually the curve will overtake the line.

The arc of the moral universe is long, but it bends towards justice. -Theodore Parker

Answer 1

Hint: If you write down the equation for the elliptic curve (at least the affine part) and the line explicitely and then solve for intersection you will find the third solution.

Answer 2

By eliminating $y$, you get a cubic polynomial. By the fundamental theorem of algebra, cubics have one or three roots (degenerate cases excluded).