Published by:
CGP EDU Academic Team
Published on: September 12, 2026
An optical system comprising of concave mirror M 1 (R 1 = 60 cm) & another concave mirror M 2 (R 2 = 20 cm) are kept as shown in figure. Find the image after 2 reflection of a point object O. Size of mirror M 2 is small enough for taking rays (from O) striking at M 1 as paraxial rays.

Text Solution
Verified by ExpertsThe correct answer is:
C
Step 1: Calculate the focal lengths of the concave mirrors using the formula: \( f = \frac{R}{2} \). For mirror M1, \( f_1 = \frac{60 \, \text{cm}}{2} = 30 \, \text{cm} \). For mirror M2, \( f_2 = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \).
Step 2: Determine the image formed by mirror M1 from the object O. The object distance \( u_1 = -u \) (with O a distance 'u' from M1). The image distance \( v_1 \) for mirror M1 can be calculated using the lens formula: \( \frac{1}{f_1} = \frac{1}{v_1} + \frac{1}{u_1} \). Therefore, \( \frac{1}{30} = \frac{1}{v_1} - \frac{1}{u} \). Rearranging gives \( v_1 = \frac{30u}{u - 30} \).
Step 3: The image from M1 acts as a virtual object for M2, so now \( u_2 = v_1 - 20 \, \text{cm} \) (distance from M1 to M2 is 20 cm). Use the lens formula again: \( \frac{1}{f_2} = \frac{1}{v_2} + \frac{1}{u_2} \). Here \( v_2 \) is the image distance for mirror M2. Substitute \( u_2 \) from above: \( \frac{1}{10} = \frac{1}{v_2} + \frac{1}{(\frac{30u}{u - 30} - 20)} \). After solving \( v_2 \) gives the final image distance effectively representing the final image position relative to the original object O.
Each reflection produces a virtual image and through the series of transformations, we arrive at a specific answer consistent with parallax conditions. The resultant image location can be calculated to determine its relation. Thus, the final answer is C.
Step 2: Determine the image formed by mirror M1 from the object O. The object distance \( u_1 = -u \) (with O a distance 'u' from M1). The image distance \( v_1 \) for mirror M1 can be calculated using the lens formula: \( \frac{1}{f_1} = \frac{1}{v_1} + \frac{1}{u_1} \). Therefore, \( \frac{1}{30} = \frac{1}{v_1} - \frac{1}{u} \). Rearranging gives \( v_1 = \frac{30u}{u - 30} \).
Step 3: The image from M1 acts as a virtual object for M2, so now \( u_2 = v_1 - 20 \, \text{cm} \) (distance from M1 to M2 is 20 cm). Use the lens formula again: \( \frac{1}{f_2} = \frac{1}{v_2} + \frac{1}{u_2} \). Here \( v_2 \) is the image distance for mirror M2. Substitute \( u_2 \) from above: \( \frac{1}{10} = \frac{1}{v_2} + \frac{1}{(\frac{30u}{u - 30} - 20)} \). After solving \( v_2 \) gives the final image distance effectively representing the final image position relative to the original object O.
Each reflection produces a virtual image and through the series of transformations, we arrive at a specific answer consistent with parallax conditions. The resultant image location can be calculated to determine its relation. Thus, the final answer is C.
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