Question

how the image is formed due to an object placed in front of a convex lens having a hole at it's center

Answer 1

A ray of light passing through the Optical Center of the lens travels straight without suffering any deviation. This holds good only in the case of a thin lens.
An incident ray parallel to the principal axis after refraction passes through the focus.
An incident ray passing through the focus of a lens emerge parallel to the principal axis after refraction.

The nature of images formed by a convex lens depends upon the distance of the object from the Optical Center of the lens. Let us now see how the image is formed by a convex lens for various positions of the object.

1. When the Object is Placed between F1 and O:

Formation of Image by a Convex Lens

The image is -

Formed on the same side of the lensVirtualErectMagnified
2. When the Object is Placed between the Optical Center (O) and first Focus (F1)

Here we consider two rays starting from the top of the object placed at F1 and optical center. The ray parallel to the principal axis after refraction passes through the focus (F2). The ray passing through the optical center goes through the lens undeviated. These refracted rays appear to meet only when produced backwards. Thus, when an object is placed between F1 and O of a convex lens, a virtual, erect and magnified image of the object is formed on the same side of the lens as the object.
3. When the Object is Placed at 2F1

The image is -

Formed at 2F2RealInvertedSame size as the object

Here one of the rays starting from the top of the object placed at 2F1 passes through the optic center without any deviation and the other ray which is parallel to the principal axis after refraction passes through the focus. These two refracted rays meet at 2F2. Thus, when an object is placed at 2F1 of a convex lens, inverted and real image of the same size as the object is formed at 2F2 on the other side of the lens.

4. When the Object is Placed between F1 and F2

The image is

Formed beyond 2F2RealInvertedMagnified

Let us consider two rays coming from the object. The ray which is parallel to the principal axis after refraction passes through the lens and passes through F2 on the other side of the lens. The ray passing through the optic center comes out of the lens without any deviation. The two refracted rays intersect each other at a point beyond 2F2. So, when an object is placed between F1 and 2F1 of a convex lens the image is formed beyond 2F2.

5. When the Object is Placed at F1

The image is -

Formed at infinityrealInvertedMagnified

Here again we consider two rays coming from the top of the object. One of the rays which is parallel to the principal axis after refraction passes through F2 and the other ray which passes through the optical center comes out without any deviation. These two refracted rays are parallel to each other and parallel rays meet only at infinity. Thus, when an object is placed at F1 of a convex lens, the image is formed at infinity and it is inverted, real and magnified.

6. When the Object is Placed beyond 2F1

The image is -

Formed between F2 and 2F2RealInvertedDiminished

The ray parallel to the principal axis after refraction passes through F2 and the ray which passes through the optical center comes out without any deviation. The refracted rays intersect at a point between F2 and 2F2. The image is inverted, real and diminished.
7. When the Object is Placed at Infinity

The image is -Formed at F2InvertedRealHighly diminished