Distinguish between lateral magnification and longitudinal magnification. Show that longitudinal magnification is equal to square of lateral magnification.

There is an object on one side and a screen on the other side of a convex lens. Show that there are two positions of the convex lens for which a real image of the object is formed on the screen if the distance between the object of the screen is more than four times the focal length of the lens.

If x be the distance between the two positions of the lens and d the distance between the object and the screen, then show that $\mathrm{f}=\left({\mathrm{d}}^2-{\mathrm{x}}^2\right)/4\mathrm{d}$.