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. `sqrt(x_(1)x_(2))`
B. `(x_(1)x_(2))/(2)`
C. `(x_(1)+x_(2))/(2)`
D. `x_(1)x_(2)`

Answer

A

Explanation

For the convex lens, `u=-(f+x_(1))`
and for the real image, `v=v+x_(2)`
`therefore` For the lens, `(1)/(v)-(1)/(u)=(1)/(f)`
`therefore (1)/(f+x_(2))+(1)/(f+x_(1))=(1)/(f)`
`therefore (f+x_(1)+f+x_(2))/((f+x_(1))(f+x_(2)))=(1)/(f)`
`therefore f(2f+x_(1)+x_(2))=f^(2)+fx_(1)+fx_(2)+x_(1)x_(2)`
`therefore 2f^(2)+fx_(1)+fx_(2)=f^(2)+fx_(1)+fx_(2)+x_(1)x_(2)`
`therefore f^(2)=x_(1)x_(2) or f= sqrt(x_(1)x_(2))`
This is Newton's formula.
[ We can directly use the formula `f=sqrt(x_(1)x_(2))`.]