An object is placed at a distance `x_(1)` from the focus of a concave mirror. Its real image is formed at a distance `x_(2)` from the focus. Show that the focal length of the mirror is `sqrt(x_(1)x_(2))`.

Explanation

Now, according to mirror equation,
`(1)/(v) + (1)/(u) = (1)/(f)` and
`rArr u = f + x_(1)` and `v = f + x_(2)`

`rArr - (1)/((f + x_(2))) - (1)/((f+x_(1))) = (1)/(-f)`
`rArr (f + x_(1) + f + x_(2))/((f + x_(2)) (f + x_(1))) = (1)/(f)`
`2f^(2) + fx_(1) + fx_(2) = f^(2) + fx_(1) + fx_(2) + x_(1)x_(2)`
`rArr f^(2) = x_(1)x_(2) rArr f = sqrt(x_(1)x_(2))`