Image formed via a converging lens when the object is placed at focal point

Question

In the case where the object is placed at focal point F of a converging lens, the rays of light (the one passing through O and the one sent parallel to the principal axis never meet) so I am assuming that the image of that object at that specific position does not exist. However, in my class notes, the teacher has said that the image is formed at infinity. I searched and then I found three answers:

1. The image indeed does not exist.
2. The image is formed at infinity.
3. In the third answer, they say that if the object is placed between the focal point and the lens but very close to the focal point, then the image is formed at infinity. They say this although they were discussing the case where the object is at F, not "very close to F from inside". However, they could not help but mention that in that case.

The weird thing about these answers is that they say that the image is a real image. How can image be real if it does not exist to start with?

I am confused. Does it exist or does it not? And what about the 3rd common answer I mentioned above? Why do some people say while explaining the case where object is on F, that "when the object is between F and lens but very close to F...'"? It is very confusing.

Answer 1

The image could be real or virtual. We'll start with a real image. Also, we'll consider a point object and an ideal lens.

For a real image of a point to be formed, the rays emitted by or reflected from that point have to converge at some other point in space.

If a point (blue dot on the diagrams below) is placed in a focal plane of a convex lens and its rays, collected by the lens, are coming out parallel to each other, they, obviously, are not going to to converge and, therefore, are not going to form an image.

If a point is placed in front of the focal plane, the rays are going to converge and form a real image.

If a point is placed behind the focal plane (i.e. between the focal plane and the lens), the rays are going to diverge and, therefore are not going to form a real image. If the diverging rays are extended backwards, they will meet at some point (of the apparent divergence) behind the lens, forming a virtual image.

Hopefully, this clarifies the picture.

Answer 2

Let the object be at a distance of $2f$ to the left of the lens where $f$ is the focal length of the lens.
A point image will be formed at a distance of $2f$ to the right of the lens and it will be real in that the image can be seen on a screen placed $2f$ to the right of the lens as the refracted rays converge and meet at the position of the image.

Now move the object closer to the lens and the image will be formed further to the right from the lens.
Continue moving the lens closer to the lens until it is just a very,very small distance to the left of the focal point. A real image is formed a very long distance away to the right lens. The refracted rays will still converge but will be close to be parallel.

Now move the object so that it is at the focal point. The emergent refracted rays will be parallel and thus will never meet. However if one assumes that parallel rays meet at infinity then for convenience one can say that the image is formed at infinity. This is situation 1.

Start with the object less than $f$ from the lens and the refracted rays will then be divergent and so never meet to the right of the lens.
However if one back produces the refracted rays they would meet at a point to the left of the lens and we would say that a virtual image is formed at that point.

Now move the object to the left further away from the lens and closer to the focal point. The refracted rays will now be less divergent and the virtual image would be formed further to the left from the lens.

Continue moving the object until it is at the focal point and now the refracted rays will be parallel and so the back produced rays will never meet, or you could say meet at infinity to the left of the lens. This is situation 2.

So you now have two situations, 1 and 2, when the object is that the focal point with images being formed at infinity to the right of the lens and at infinity to the left of the lens. Calling one image real and the other virtual might help one decide which of the two ray constructions you are using?