A short object of length `L` is placed along the principal axis of a concave mirror away from focus. The object distance is `u`. If the mirror has a focal length `f`, what will be the length of the image ? You may take `L lt lt | u - f |`.

Explanation

Let the two ends of the object be at distances `u_(1) = (u = L//2)` and `u_(2) = (u + L//2)` from the pole of concave mirror, so that `|u_(1) - u_(2)| = L = `size of the object.
Let the image of the two ends be formed respectively at `v_(1)` and `v_(2)` on reflection from the concave mirror.
`:.` Size of image `= L' = |v_(1) - v_(2)|`
From mirror formula, `(1)/(v) + (1)/(u) = (1)/(f)`
`(1)/(v) = (1)/(f) - (1)/(u) = (u - f)/(fu) or v = (f u)/(u - f)`
`v_(1) = (fu_(1))/(u_(1) - f) = (f(u - L//2))/(u - f - L//2) and v_(2) = (fu_(2))/(u_(2) - f) = (f(u + L//2))/(u - f + L//2)`
`L' = |v_(2) - v_(2)| = (f(u - L//2))/((u - f - L//2)) - (f(u + L//2))/((u - f + L//2))`
`= ((u - f)(fu - (f L)/(2) - f u - (f L)/(2)) + (L)/(2)(fu - (f L)/(2) + (f L)/(2)))/((u - f)^(2) - L^(2)//4)`
`L' = ((u - 1)(- f L) + (L)/(2). 2 fu)/((u - 1)^(2) - L^(2)//4) = (f^(2) L)/((u - f)^(2) - L^(2)//4)`
As the object is short and kept away from focus, therefore, `L^(2)//4 lt lt (u - f)^(2)`.
Hence, `L' = (f^(2)L)/((u - f)^(2))`