A small linear object of length I is placed along principal axis of a concave mirror. The neartest point the object is at a distance d from the mirror. If focal length of the mirrors is f then, prove that the size (length) of the image will be `l((f)/(d-f))^(2)`

Explanation

Equation of a concave mirror,
`(1)/(v) + (1)/(u) = (1)/(f)`
Now differentating with respect to u,
`-(1)/(v^(2))(dv)/(du) - (1)/(u^(2)) = 0 or, (dv)/(du) = - (v^(2))/(u^(2))`
Therefore, magnitude of axial magnification of the image
`m = |dv|/|du| = ((v)/(u))^(2)`
From equation of mirror we get, `(u)/(v) + 1 = (u)/(f)`
`or, " " (u)/(v) = (u-f)/(f) = (d-f)/(f)` [`therefore` given, u = d]
`therefore " " m=((v)/(u))^(2) = ((d-f)/(f))^(2)`
`therefore "Lenght of the image" = l xx m = l ((f)/(d-f))^(2).`