Question

An object is X, focal length to left of principal focus of a convex mirror. Show that the image will be 1/X , of focal length to left of the focus?

Answer 1

Here, focal length of convex mirror = f

Let object is placed at left of mirror .

As, object is X focal length to left of principal focus

Thus, object distance , u = Xf - f

                                        = f *(X - 1)

Applying mirror formula -

1/v + 1/u = 1/f

where, v = image distance

           u = object distance  =  - f *(X - 1)

            f = focal length

=> 1/v - 1/[f *(X - 1)]  =  1/f

=>  1/v =  1/f + 1/[f *(X - 1)]

=>  1/v  =  (X -1 + 1)/[f *(X - 1)]

=>  1/v =  X/[f *(X - 1)]

=>  v  =  [f *(X - 1)]/X

Thus, image distance  =  [f *(X - 1)]/X

For calculating , the distance of image from focus -

v'' = f - v

   = f  -   [f *(X - 1)]/X

    = (fX - fX + f)/X

     =  f/X

Thus,  v''  =  f * (1/X)

So, image will be 1/X , of focal length to left of the focus .