One side of radius of curvature `R_2=120 cm` of a convexo-convex lens of material of refractive index `mu=1.5` and focal length `f_1=40 cm` is slivered. It is placed on a horizontal surface with silvered surface in contact with it. Another convex lens of focal length `f_2=20 cm` is fixed coaxially `d=10 cm` above the first lens. A luminous point object O on the axis gives rise to an image coincide with it. Find its height above the upper lens.

Explanation

The lower lens will behave as a concave mirror of focal length `F`
`(1)/(F)=(1)/(f_l)+(1)/((R_2)/(2))+(1)/(f_l)=(2)/(f_l)+(2)/(R_2)`
`=(2)/(40)+(2)/(120)=(1)/(20)+(1)/(60)=(4)/(60)=(1)/(15)`
`F=15cm`
If by object and mirror combination object and image are formed at same place, for this image by upper lens should be formed at centre of curvature of mirror.
For upper lens:
`u=-x,v=-(30-10)=-20cm,f=20cm`
`(1)/(v)-(1)/(u)=(1)/(f)`
`(1)/(-20)-(1)/(-x)=(1)/(20)implies(1)/(x)=(1)/(20)+(1)/(20)=(1)/(10)`
`x=10cm`