Suppose there is a lens and a mirror. An object is placed in front of this system (nearer to the lens). If the final image of object (refraction, reflection and again refraction) is formed on the object itself, then must the each ray falling on the mirror be perpendicular to the mirror so that it can retrace its path? Is there no other situation except this in which the image of the object will form on the object itself?
The rays don't have to be perpendicular to the mirror for this to happen. Here is an example.
Let the lens be a converging lens of focal length $f$. If the object is a distance $2f$ away from the lens, and the lens is a distance $2f$ away from the mirror:
- The image of the object through the lens is co-located with the mirror, inverted and real. Call this image 1.
- The image of image 1 through the mirror coincides with image 1, and is inverted (relative to the object orientation). Call this image 2.
- The image of image 2 through the lens is co-located with the object, upright (relative to the object) and real. Therefore it coincides with the object.
The rays forming image 1 and reflecting off the mirror are not all perpendicular to the mirror. What's happening here is each ray not retracing its own path, but that of another ray emitted from the same point in the object. So each ray transmitted through the optical system ends up at the point it was emitted from.