A convex lens of focal length $20\ \text{cm}$ is placed in front of a convex mirror with principal axis coinciding with each other. The distance between the lens and mirror is $10\ \text{cm}$. A point object is placed on the principal axis at a distance of $60\ \text{cm}$ from the convex lens. The image formed by the combination coincides with the object itself. The focal length of the convex mirror is ______ cm.
Numerical value type. Enter your answer.
Answer: 10
Lens: $\dfrac1v = \dfrac1{20} - \dfrac1{60} = \dfrac1{30}\Rightarrow v = 30\ \text{cm}$, i.e. $20\ \text{cm}$ behind the mirror.
For the light to retrace its path the rays must hit the mirror normally, so they converge towards its centre of curvature: $R = 20\ \text{cm}$ and $f = \dfrac R2 = 10\ \text{cm}$.
Solution by Sreeraj P, M.Sc Physics