An object is placed at a distance `x_1` from the principal focus of a lens and its real image is formed at a distance `x_2` from the another principal focus. The focal length of the lens is

Options

A. `x_1x_2`
B. `(x_1x_2)/(2)`
C. `(x_1+x_2)/(2)`
D. `sqrt9x_1x_2)`

Answer

D

Explanation

`u=-(f+x_1),v=(f+x_2)`
`(1)/(f)=(1)/(v)-(1)/(u)=(1)/(f+x_2)-(1)/(-(f+x_1))`
`(1)/(f+x_2)+(1)/(f+x_1)=(2f+x_1+x_2)/((f+x_2)(f+x_1))`
`(f+x_2)(f+x_1)=f[2f+x_1+x_2]`
`f^2+f(x_1+x_2)+x_1x_2=2f^2+f(x_1+x_2)`
`f^2=x_1x_2` i.e., `f=sqrt(x_1x_2)`