In the following figure two mirrors `M_(1) and M_(2)` are placed in front of each others.

Both the mirrors have same focal length of 30 cm. Mirror `M_(1)` is concave but mirror `M_(2)` is convex. Distance between the two mirrors is d and their principal axes are coinciding as shown in figure. An object O is placed on principal axis at a distance 45 cm from mirror `M_(1)`. It is found that final image is formed on object itself. Calculate the value of d.

Explanation

First we can solve for reflection through mirror `M_1`.
`1/f = 1/v + 1/u`
`1/(-30) = 1/v + 1/(-45)`
`implies 1/v= 1/(45) -1/(30)`
`implies 1/v = (2-2)/(90)`
`implies v=-90` cm
This image will serve the purpose of object for convex mirror. If the image formed by `M_2` has to coincide with Othen object for convex mirror must be virtual so that it may form real image on O. Hence `d lt 90` cm .

Now we can apply mirror formula for convex mirror.
`1/f = 1/v + 1/u`
`implies 1/(30) =1/(-(d-45)) + 1/(+(90-d))`
`implies 1/(30) = (1)/((90-d)) -(1)/((d-45))`
`implies 1/(30) = ((d-45)-(90-d))/((90-d) (d-45))`
`implies 1/(30) = (2d-135)/(90d-4050 -d^2 + 45d)`