Where is the image formed by a concave lens when the object is between optical centre O and infinity?

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Object is at infinity:
Object is between optical centre, O and infinity:
Sign convention for lens:

In a concave lens, when an object is between infinity and the optical centre O, an image is formed between the .

focus (F $_{1}$ ) and optical centre (O)

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Object at Infinity
Object Between O and Infinity
Sign Convention For Lens

There are only two possibilities of position of object in the case of a concave lens:

Object at Infinity	Image is formed on the same side as object with respect to lens. Image is virtual, erect and point sized.
Between O and inifinity	Image is formed on the same side as object with respect to lens. Image is virtual, erect and smaller than object.

In each of these cases, image is highly reduced in size, is erect and virtual and formed in front of the lens, i.e. on the same size as object.

Object is at infinity:

A highly diminished point sized, virtual and erect image is formed when object is at infinity by a concave lens at principal focus F1.

Fig: Object at Infinity

This figure shows a concave lens. Light rays are parallel to the principal axis which shows that they are coming from infinity. After passing through the lens, light rays diverge from the lens and appear to be coming from the focus. That is why image is formed on the focus of the lens.

Properties of Image: Image is point sized, highly diminished, virtual and erect.

Object is between optical centre, O and infinity:

A diminished, virtual and erect image is formed between principal focus F1 and optical centre, O; when object is placed between optical centre and infinity of a concave lens.

Fig: Object between Infinity and O

This figure shows a concave lens. One of the rays of light is parallel to the principal axis and another ray of light is passing through the optical centre of lens. When both the emergent rays are drawn backwards they appear to be intersecting somewhere between F and O. So, image if formed between F and O.

Properties of Image: Image is diminished, virtual and erect.

Sign convention for lens:

Sign convention for lens is similar to that of spherical mirror. Signs are taken left of the optical centre as negative, right of the optical centre as positive, above of the principal axis as positive and below of the principal axis as negative.

Fig: Sign Convention

The new sign convention is known as New Cartesian Sign Convention. In this sign is taken negative towards left and taken as positive towards right at X-axis from origin.

The sign is taken as positive (+) above the origin point at Y-axis; and below the origin point as negative (-) at Y-axis.