A combination of two thin lenses with focal lengths `f_(1)` and `f_(2)` respectively forms and image of distant object at distance `60cm` when lenses are in contact. The position of this image shifts by `30 cm` towards the combination when two lenses are separated by `10 cm`. The corresponding values of `f_(1)` and `f_(2)` are

Options

A. 30 cm, –60 cm
B. 20 cm, –30 cm
C. 15 cm, –20 cm
D. 12 cm, –15 cm

Answer

B

Explanation

Case - I :
`(1)/(f_(eq)) = (1)/(f_(1)) + (1)/(f_(2))`
`rArr " " (1)/(60) = (1)/(f_(1)) + (1)/(f_(2))" "`….(i)
Case- II :
`(1)/(f_(eq)) = (1)/(f_(1)) + (1)/(f_(2)) - (d)/(f_(1) f_(2))`
`rArr " " (1)/(30) = (1)/(f_(1)) + (1)/(f_(2)) - (10)/(f_(1) f_(2))" "`..... (ii)
`rArr " " (1)/(30) = (1)/(60) - (10)/(f_(1) f_(2)) rArr (10)/(f_(1) f_(2)) = (1)/(60) - (1)/(30) = (1- 2)/(60)`
`rArr f_(1) f_(2) =- 600 " " ` .... (iii)
From eq. (i) `(1)/(60) = (f_(1) + f_(2))/(f_(1) f_(2)) rArr (1)/(60) = (f_(1) + f_(2))/(- 600)`
`rArr " " f_(1) + f_(2) = - 10 " "`... (iv)
Now , `(f_(1) - f_(1))^(2) = (f_(1) + f_(2))^(2) - 4f_(1) f_(2)`
`rArr " " (f_(1) - f_(2))^(2) = 100 - 4 (-600) rArr (f_(1) - f_(2))^(2) = 2500`
`rArr " " f_(1) - f_(2) = 50" "`...(v)
From eqn. (iv) & (v)
`2f_(1) = 40 rArr f_(1) = 20` cm
and `f_(2) = - 30` cm .