An object is placed at a distance of 12 cm from a convex lens. A convex mirror of focal length 15 cm is placed on another side of the lens at 8 cm as shown in the figure. The image of the object coincides with the object.

When the convex mirror is removed, a real and inverted image is formed at a position. The distance of the image from the object will be _________ cm

We need to find the total distance between the object and its final real image when the convex mirror is removed from the system.

1. Analyze the First Case (With the Convex Mirror)

We are given that the final image of the object coincides with the object itself. This means light rays traveling from the object pass through the convex lens, strike the convex mirror, and trace their paths exactly back to the object.

For light rays to retrace their path after reflecting off a convex mirror, they must hit the mirror surface normally (perpendicularly). This happens only if the refracted rays from the lens are traveling toward the center of curvature ($C$) of the convex mirror.

• Focal length of the convex mirror ($f_m$): $$15\text{ cm}$$
• Radius of curvature of the mirror ($R$):

  $$R = 2 \times f_m = 2 \times 15\text{ cm} = 30\text{ cm}$$

Therefore, in the absence of the mirror, the lens would form a virtual image at the center of curvature of the mirror. Let's calculate this position ($v$) relative to the lens:

$$\text{Distance from lens to mirror} = 8\text{ cm}$$

$$\text{Distance from mirror to } C = 30\text{ cm}$$

$$v = 8\text{ cm} + 30\text{ cm} = 38\text{ cm}$$

So, the image distance for the lens is $$v = +38\text{ cm}$$.

2. Analyze the Second Case (Mirror is Removed)

When the convex mirror is removed, the light rays simply continue along their path unobstructed and converge to form a real image at that very same position ($v = 38\text{ cm}$) from the lens.

Let's map out the positions of the key elements along the principal axis:

• The object is placed at a distance of $$12\text{ cm}$$ to the left of the lens.
• The real image is formed at a distance of $$38\text{ cm}$$ to the right of the lens.

3. Calculate the Distance Between Object and Image

The total horizontal distance ($D$) from the object to its final real image is the sum of the object distance and the image distance relative to the lens:

$$D = \text{Distance from Object to Lens} + \text{Distance from Lens to Image}$$

$$D = 12\text{ cm} + 38\text{ cm} = 50\text{ cm}$$

Final Answer: 50