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.