If the distance D between an object and screen than 4 times the focal length of a convex lens, then there are two positions of the lens for which image are formed on the screen. This method is called conjugate method. If d is the distance between the two positions of the lens, obtain the equation for focal length of the convex lens.

Explanation

Mirror `1/f = 1/v + 1/u`
Lens us eliminate v by substituting v = D- u
` (1)/(D - u) + 1/u - 1/f`
We get the equation `u^2 - Du + fD = 0`
Quaratic equation ` u = ( D pm sqrt(D^2 - 4fD))/(2)`
` mu_1 = (D - sqrt(D^2 - 4fD))/(2)`
`v_1 = D - u_1 = (D + sqrt(D^2 - 4fD))/(2)`
after changing
`mu_1 = (D + sqrt(D^2 - 4f D))/(2)`
` v_2 = D - u_2 = (D - sqrt(D^2 - 4fD))/(2)`
now the displacement
` d = v_1 - u_1 - u_2 - v_2 = sqrt(D^2 - 4fD)`
` f = (D^2 - d^2)/(4D)`