A point object is placed at a distance of 12 cm on the axis of a convex lens of focal length 10 cm. On the other side of the lens, a convex mirror is placed at a distance of 10 cm from the lens such that the image formed by the combination coincides with the object itself. What is the focal length of the convex mirror?

The correct option is D

25 cm

The incident ray OA is refracted along BD by the lens. Since the image coincides with the object, the refracted ray BD must retrace its path after reflection from the mirror, i.e., it falls normally on the mirror. Let C be the point on the axis where BD produced meets the axis. It is clear that C is the centre of curvature of the mirror. Also C is the image of the object O formed by the lens alone.

Thus
f = + 10 cm and u = –12 cm

The distance v of the image is obtained from the lens formula
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

Which gives v = –60 cm

Since x = 10 cm, R = v – x = 60 – 10 = 50 cm

Hence the focal length of the mirror $=\frac{R}{2}=25cm$.