The distance between two point object P and Q is 32 cm, a convex lens of focal length 15 cm is palced bewteen them so that the images of both the objects are formed at the same place. The distance of P from the lens could be
(i) 20 cm
(ii) 18 cm
(iii) 16 cm
(iv) 12 cm

Options

A. (i),(ii)
B. (ii),(iii)
C. (iii),(iv)
D. (i),(iv)

Answer

D

Explanation

Since both images are formd at the same place, one image is real another virtual.
Let image of `O_1` is real.
image of `O_1`: `(1)/(v_1)-(1)/(-x)=(1)/(15)implies(1)/(v_1)=(1)/(15)-(1)/(x)`
Image of `O_2`: `(1)/(v_2)-(1)/(-(32-x))=(1)/(15)implies(1)/(v_2)=(1)/(15)-(1)/((32-x))`
`v_1` is `+ve` is `-ve`
`v_1=-v_2`
`(1)/(15)-(1)/(x)=-((1)/(15)-(1)/(32-x))`
`(1)/(x)+(1)/(32-x)=(2)/(15)implies(32)/(x(32-x))=(2)/(15)`
`(16)/(x(32-x))=(1)/(15)`
`240=32x-x^2`
`x^2-32x+240=0`
`(x-20)(x-12)=0impliesx=20cm,12cm`