Suppose I be the real image of an object at O. Let d represent the distance between them. If the image is x distance from the lens’s centre, then the distance between the object and the lens’s centre is (dx).
As we know, sign convention for convex lenses.
Thus, u=−(d−x) and v=+x
1/f = 1/x − 1/−(d−x)
⇒ 1/f = 1/x + 1/(d−x)
⇒ x2 − xd − fd = 0
For a real image, the value of x must be a real number, for roots of equation will be real
when, d2 ⩾ 4fd
d⩾4f
Hence, from the above equation we get the minimum distance between an object and its real image formed by a convex lens is
d=4f .