dear sir this was posted long back by my friend to me i gave him the solution but he said that was wrong ; i gave him answer as (B) and (C) this is multiple choice question

Dear student,

The eccentricity of an ellipse is defined as the ratio of the distance between any point in the ellipse and the focus and distance between point and a fixed line called directrix.

            e = distance between P and F/distance between P and line D (Directrix)

Formula for eccentricity :      

               Formula for calculating e is

                b² = (a²)(e²-1)    So e²- 1 = b²/a²     or e² = b²/a² +1 

                e = √(b²/a²+1)

                Eccenticity = square root of square of semi minor axis divided by squares of semi major axis

eccentricity of hyperbola = 1+b²/a² = e         ..............1

eccentricity of conjugate = 1+a²/b² = e_c²        ..................2    (e_c = eccentricity of conjugate hyperbola)

from these two equatns , e_c  = e/(e²-1)^1/2            ......................3

f(f(e)) = e                          (n = 2 even)

f(f(f(e))) = e/(e²-1)^1/2       (n = 3 odd)

now , if n = even then

integral ede = e²/2 lim from 1 to 3

                  =4                                   (A)

if n = odd then

 integral  ede/(e²-1)^1/2  lim 1 to 3

      = (e²-1)^1/2          lim 1 to 3

       =2root2                                     (D)

hence option A,D are correct ....

approve if u like my ans

Approved