A diverging lens of focal length 20 cm and a diverging mirror of focal length 7.5 cm are placed 5.0 cm apart with their principal axes coinciding. Where should an object be placed so that its image falls on itself?

Let the object to place at a distance x from the lens further away from the mirror.

As shown in the figure, for the concave lens (1st refraction)

using lens formula,

So, the virtual image due to fist refraction lies on the same side as that of object. (A’B’) This image act as the object for the concave mirror. For the mirror,

Also it is given that the focal length, f= -10cm. Hence, from the mirror equation,

So, this image is formed towards left of the mirror. Again, for second refraction in concave lens, hence, assuming that image of mirror is formed between the lens and mirror

Since, the final image is produced on the object

Using lens formula,

The object should be placed at a distance 60 cm from the lens further away from the mirror. So that the final image is formed on itself.