Let us take two convex lenses made up of the same material so that their refractive index (n) is same.
i) The first case given is two lenses are touching. So let us define the lens as lens A & lens B and their focal length be f1 & f2 respectively.
Now place a point object in front of the two lenses at a distance of u. Rays coming from the object after refraction from lens A forms an image I1 at a distance v1. Now this I1 will act as an object for lens B and after refraction from the lenses will form an image I at distance v.
Using lens formula:
Lens A: ….(1)
Lens B: 1/v-1/v1 = 1/f2 …(2)
Adding equation (1) & (2) we get,
Where F = effective focal length of lens
ii) Lenses are separated by a distance 'a' with common principle axis.
Let O be the point object placed at a distance u1. Rays from it after refraction from lens 1 alone will converge at I1 at a distance v1. But lens 2 will converge it before I1 creating the actual image I t distance v.
Object distance for lens 1 = u1 = OC1
where C1 is center of curvature of lens 1
Image distance for lens 1 = v1 = I1C1
Object distance for lens 2 = I1C2 = u2
Image distance for lens 2 = IC2 = v2
where C2 is center of curvature of lens 2
Now from the figure: I1C1 = I1C1 + a
using lens formula:
Lens 1: = ….(1)
Solving ,, & 
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