A convex lens, of focal length 30 cm, a concave lens of focal length 120 cm, and a plane mirror are arranged as shown. For an object kept at a distance of 60 cm from the convex lens, the final image, formed by the combination, is a real image, at a distance of:

The lens formula is $$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$

1. Reflection from convex lens:

$$\frac{1}{v_1} - \frac{1}{-60} = \frac{1}{30} \implies \frac{1}{v_1} = \frac{1}{30} - \frac{1}{60} = \frac{1}{60}$$

$$v_1 = +60$$ cm

2. Reflection from concave lens:

The concave lens is placed $$20$$ cm to the right of the convex lens. The first image acts as a virtual object for the concave lens.

Object distance ($$u_2$$): $$60\text{ cm} - 20\text{ cm} = +40$$ cm.

$$\frac{1}{v_2} - \frac{1}{40} = \frac{1}{-120} \implies \frac{1}{v_2} = \frac{1}{40} - \frac{1}{120} = \frac{3-1}{120} = \frac{1}{60}$$

$$v_2 = +60$$ cm

3. Reflection from plane mirror:

The distance between the convex lens and the mirror is $$70$$ cm. The distance between the concave lens and the mirror is $$70\text{ cm} - 20\text{ cm} = 50$$ cm. The image from the concave lens is $$60$$ cm to its right, which puts it $$60\text{ cm} - 50\text{ cm} = 10$$ cm behind the mirror. For a plane mirror, a virtual object $$10$$ cm behind it produces a real image $$10$$ cm in front of it.

4. Final image:

The mirror is located at $$70$$ cm from the convex lens. The real image is $$10$$ cm in front (to the left) of the mirror.

Distance from convex lens = $$70\text{ cm} - 10\text{ cm} = 60$$ cm.