Question #74b2b

We know the conjugate foci relation of lens as follows

`color(blue)(1/v-1/u=1/f)`

Where

`u->"object distance"`

`v->"image distance"`

`f->"focal length"`

Imposing sign convention

`u=-u" for real object"`

`v=+v" for real image"`

`f=+f" for converging lens"`

So the lens formula becomes

`color(green)(1/v-1/(-u)=1/f)`

`=>color(green)(1/v+1/u=1/f)`

Now let us consider that the distance between real object and real image be `D`. This means `u+v=D`. So putting `v=D-u` in the lens equation we get

`color(red)(1/(D-u)+1/u=1/f)`

`color(red)(=>(u+D-u)/(u(D-u))=1/f)`

`color(red)(=>Df=(u(D-u)))`

`color(red)(=>u^2-Du+Df=0)`

This is a quadratic equation of `u`. For the formation of real images for real object the roots of this equation should be real.To satisfy this condition the discriminant of this equation must be `>=0`

So `color(violet)(D^2-4Df>=0)`

`color(violet)(=>D^2>=4Df)`

`color(violet)(=>D>=4f)`

From this relation we can conclude that the minimum distance between real obejct and its real image formed by a converging lens is `4f`

The following graph supports the phenomenon discussed above.