A converging lens having a focal length of 15 cm and a converging...

Let the point source be placed at a distance $x$ from the lens.

The distance of the point source from the mirror is $50 - x$.

Two images are formed by the optical system for an observer looking through the lens: one by the direct refraction of light through the lens, and another by the reflection of light from the mirror followed by refraction through the lens.

For these two final images to form at the identical position, the light in both paths must act as identical objects for the lens. This means the intermediate image formed by the mirror must coincide exactly with the position of the point source.

A converging mirror forms an image at the object's location when the object is placed at its center of curvature.

The radius of curvature of the mirror is $R = 2f = 2 \times 10 = 20$ cm.

Thus, the point source must be placed at a distance of $20$ cm from the mirror.

The location of the point source from the lens is $50 - 20 = 30$ cm.