Image distance / focal length relation

[jmcgraw]

Let O be the object, L be the lense, I be the image, and S be a screen for projecting an image. The lens can be placed anywhere between O and S. Let f be the focal length of the lense. Prove that for an image to be clearly formed on screen S, OS > 4f must be true.

A figure of a "too short" OS distance (since the image would be past the screen) would look like this:

O-----L-----------S----I


I am having a very hard time proving that OS>4f must be true! Any hints?

 

[andrevdh]

For the image distance d_i subs s-d_o where s is the distance between object and screen. After a bit of work you should get to
s=\frac{{d^2}_o}{d_o-f}
which gives you a quadratic equation in d_o
By requiring that the determinant should be positve the result follows.