A finite size object is placed normal to the principal axis at a distance of $30\ \text{cm}$ from a convex mirror of focal length $30\ \text{cm}$. A plane mirror is now placed in such a way that the images produced by both the mirrors coincide with each other. The distance between the two mirrors is:
Answer: (B) $7.5\ \text{cm}$
Convex mirror: $f = +30\ \text{cm}$, $u = -30\ \text{cm}$:
$$\frac1v = \frac1f - \frac1u = \frac1{30} + \frac1{30} \Rightarrow v = +15\ \text{cm}$$
The image is $15\ \text{cm}$ behind the convex mirror, i.e. $45\ \text{cm}$ from the object. A plane mirror forms its image as far behind it as the object is in front, so to coincide it must be halfway, $22.5\ \text{cm}$ from the object (between the object and the convex mirror, covering only part of the beam).
Distance between the mirrors: $30 - 22.5 = 7.5\ \text{cm}$.
Solution by Sreeraj P, M.Sc Physics