A convex lens of focal length $$20 \text{ cm}$$ is placed in front of convex mirror with principal axis coinciding each other. The distance between the lens and mirror is $$10 \text{ cm}$$. A point object is placed on principal axis at a distance of $$60 \text{ cm}$$ from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is ______ cm.

We need to determine the focal length of the convex mirror when it is placed in combination with a convex lens such that the final image coincides with the point object itself.

1. Analyze the First Refraction (Through the Convex Lens)

Using the lens formula, we find where the convex lens forms the first image:

$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f_{\text{lens}}}$$

From the problem

• Focal length of the convex lens ($$f_{\text{lens}}$$) = $$20\text{ cm}$$
• Object distance ($$u$$) = $$-60\text{ cm}$$ (using standard Cartesian sign convention)

Substitute these values to solve for the image distance ($$v$$):

$$\frac{1}{v} - \frac{1}{-60} = \frac{1}{20}$$

$$\frac{1}{v} + \frac{1}{60} = \frac{1}{20}$$

$$\frac{1}{v} = \frac{1}{20} - \frac{1}{60} = \frac{3 - 1}{60} = \frac{2}{60} = \frac{1}{30}$$

$$v = 30\text{ cm}$$

This means the lens forms a real image $$30\text{ cm}$$ behind it.

2. Analyze the Reflection (By the Convex Mirror)

The problem states that the final image retraces its path and coincides with the object itself. For a light ray to retrace its path after hitting a spherical mirror, it must strike the mirror surface normally (perpendicularly).

This happens only if the rays heading from the lens are directed straight toward the center of curvature ($$C$$) of the convex mirror.

• The distance from the lens to the image position is $$30\text{ cm}$$.
• The distance between the lens and the mirror is given as $$10\text{ cm}$$.

Therefore, the distance from the mirror to its center of curvature ($$R$$) is:

$$R = 30\text{ cm} - 10\text{ cm} = 20\text{ cm}$$

3. Calculate the Focal Length of the Mirror ($$f_{\text{mirror}}$$)

The relationship between the focal length and the radius of curvature for a spherical mirror is:

$$f_{\text{mirror}} = \frac{R}{2}$$

$$f_{\text{mirror}} = \frac{20}{2} = 10\text{ cm}$$

Conclusion

The focal length of the convex mirror is 10 cm.