A convex lens of focal length `30 cm` and a concave lens of focal length `60 cm` are placed in combination. If an object is placed `40 cm` away from the combination, find the position of the image.

Explanation

We shall start with refraction from concave lens . We can use lens formula to locate the image as follows :
`(1)/(v) - (1)/(u) = (1)/(f)`
`implies (1)/(v) - (1)/(- 10) = (1)/(-20)`
`implies (1)/(v) = - (1)/(10) - (1)/(20) implies v = - (20)/(3)` cm
Hence image formed by concave lens is at a distance 20/3 cm from it on the same side as that of the object . Distance of this image from concave mirror is object distance for its reflection .
Distance of image formed by concave lens from the concave mirror `= 15 + (20)/(3) = (65)/(3)` cm
Now we can use mirror formula for the image formed by concave mirror .
`(1)/(v) + (1)/(u) = (1)/(f)`
`implies (1)/(v) + (1)/((-(65)/(3))) = (1)/((-13))`
`implies (1)/(v) - (3)/(65) = - (1)/(13)`
`implies (1)/(v) = (3)/(65) - (1)/(13) = - (2)/(65)`
`implies " " v = -32.5` cm
Hence image formed by concave mirror is at a distance 32.5 cm in front of it . This point lies ahead of concave lens . Hence this point will serve as a virtual object for refraction through concave lens again. Distance of this virtual object from concave lens = `32.5 - 15 = 17.5` cm . Remember distance of virtual object in lens formula to find the location of final image as follows :
`(1)/(v) - (1)/(u) = (1)/(f)`
`implies " " (1)/(v) - (1)/(17.5) = (1)/(-20)`
`implies " " (1)/(v) = (1)/(17.5) - (1)/(20)`
`implies " " v = (20 xx 17.5)/(20 - 17.5) = (350)/(2.5) = 140` cm .
Hence final image is formed at a distance 140 cm from the concave lens and 155 cm from concave mirror and final image will be real .