A point object is placed at a distance of 60 cm from a convex lens of focal length 30 cm. If a plane mirror we

Question

A point object is placed at a distance of 60 cm from a convex lens of focal length 30 cm. If a plane mirror were put perpendicular to the principal axis of the lens and at a distance of 40 cm from it, the final image would be formed at a distance of:

Accepted Answer

1. **Image formation by Lens:** Using the lens formula $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ with $u = -60	ext{ cm}$ and $f = +30	ext{ cm}$:
   $$ \frac{1}{v} - \frac{1}{-60} = \frac{1}{30} \implies \frac{1}{v} = \frac{1}{30} - \frac{1}{60} = \frac{1}{60} $$
   $$ v = +60	ext{ cm} $$
   In the absence of the mirror, a real image ($I_1$) would form 60 cm to the right of the lens.
2. **Interaction with Plane Mirror:** The mirror is placed 40 cm to the right of the lens. The rays converging towards $I_1$ (at 60 cm) are intercepted by the mirror. 
   - The image $I_1$ acts as a **virtual object** for the plane mirror.
   - Distance of virtual object from mirror ($u_m$) = $60	ext{ cm} (I_1) - 40	ext{ cm} (Mirror) = 20	ext{ cm}$ (behind the mirror).
3. **Final Image Formation:** A plane mirror forms an image of a virtual object at the same distance in front of it. 
   - Image distance from mirror ($v_m$) = 20 cm (in front of the mirror).
   - Since the reflected rays actually converge at this point, the final image ($I_2$) is **Real**.
4. **Position relative to Lens:** The image is formed 20 cm in front of the mirror (towards the lens). Since the mirror is 40 cm from the lens, the distance of the image from the lens is $40	ext{ cm} - 20	ext{ cm} = 20	ext{ cm}$.
5. **Conclusion:** The final image is formed 20 cm from the lens (or 20 cm from the mirror) and is a real image. Option 3 correctly describes both position and nature.

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