An equiconvex lens of refractive index `mu_(2)` is placed such that the refractive index of the surrounding media is as shown :

Explanation

Let a point object O be situated on the principal axis of the concave spherical refracting surface. Let P be pole and C be centre of curvature and PC = R be radius of curvature. Let `mu_(1)` and `mu_(2)` be refractive indices of rarer and denser medium respectively as shown in the figure.
Let incident ray OA refracts at A and bends towards the normal along AQ and when produced in backward direction, it meets at point I. So I will be the virtual image of point O.

From Snell.s law for small i and r,
`(sin i)/(sin r)=(i)/(r)=(mu_(2))/(mu_(1))`
or `mu_(1)i=mu_(2)r` ...(i)
`"In " DeltaAOC," " "In" Delta IC,`
`gamma = i + alpha " " gamma= r+beta`
or `i = gamma - alpha" or " r= gamma - beta`
Putting the values of i and r in Eq. (1), we get
`mu_(1)(gamma-alpha)=mu_(2)(gamma-beta)`
or `mu_(1)gamma-mu_(1)alpha=mu_(2)gamma-mu_(2)beta`
or `-mu_(1)alpha+mu_(2)beta=(mu_(2)-mu_(1))gamma`
Since `alpha, beta` and `gamma` are small, so they can be replaced by their tangents.
`-mu_(1) tan alpha+mu_(2)tan beta=(mu_(2)-mu_(1))tan gamma`
or `-mu_(1)(AM)/(MO)+mu_(2)(AM)/(MI)=(mu_(2)-mu_(1))(AN)/(MC)`
or `-(mu_(1))/(MO)+(mu_(2))/(MI)=(mu_(2)-mu_(1))/(MC)`
Since M is very close to P, so
`MO ~= PO, MI ~= PI` and `MC ~= PC`
`:. -(mu_(1))/(PO)+(mu_(2))/(PI)=(mu_(2)-mu_(1))/(PC)`
Using sign conventions,
PO = - u, PI = - v, PC = R, we get
`-(mu_(1))/(-u)+(mu_(2))/(v)=(mu_(2)-mu_(1))/(R)`
or `-(mu_(1))/(u)+(mu_(2))/(v)=(mu_(2)-mu_(1))/(R)`
or `(mu_(2))/(v)-(mu_(1))/(u)=(mu_(2)-mu_(1))/(R)`