## Question

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?

### Solution

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

Which gives *v* = –60 cm

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

Hence the focal length of the mirror

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