An object is placed at a distance of 15 cm from the convex lens of focal length 10 cm . on the other side of the lens , a convex mirror is placed such that its distance from the lens equals the focal length of the lens. The image formed by this combination is observed to coincide with the object itself. find the focal length of convex mirror..

Consider the following figure

Here point I would be the center of curvature of the mirror, so the focal length of the mirror would be

f = MI/2

so, we know that for lens

1/f = 1/v - 1/u

or

1/v = 1/f + 1/u

given

f = +10 cm

u = -15 cm

so,

1/v = 1/10 + 1/-15

= (-2 + 3)/30

thus, image distance

v = +30 cm

so,

MI = LI - LM

= 30 - 10

or

MI = +20 cm

thus, focal length of mirror

f = MI/2 = 20cm / 2

or

f = 10 cm