A screen is placed 90 cm away from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20 cm. Find the focal length of the lens.

Hint: We will calculate the focal length of this configuration using a simple formula that relates the focal length of the lens with the distance of the screen with the object and the distance between the two locations of the lens which we’ve all been given.
Formula used: In this solution, we will use the following formula:
$f = \dfrac{{{D^2} - {d^2}}}{{4D}}$ where $f$ is the focal length of the lens in question, $D$ is the distance between the object and the screen, $d$ is the location between the two positions of the lens.

Complete step by step answer:
We have been given the distance between the object and the screen $(D = 90\,cm)$ along with the distance between the two locations of the lenses $(d = 20\,cm)$.
Then using the formula
$f = \dfrac{{{D^2} - {d^2}}}{{4D}}$, we can find the focal length of the lens as
$f = \dfrac{{{{90}^2} - {{20}^2}}}{{4 \times 90}}$
After simplifying the numerator and the denominator, we can write,
$f = \dfrac{{8100 - 400}}{{360}}$
$ \Rightarrow f = \dfrac{{7700}}{{360}} = 21.4\,cm$

Hence the focal length of the lens will be $f = 21.4\,cm$

Note: We should be aware of the formula used to calculate the focal length of such configurations as it is of wide practical use. We should also be aware of the concepts of the focal length of a lens. It is the distance at which parallel rays of light will converge or diverge by a lens. The focal length of the lens is positive since it is convex in nature. In this configuration, multiple images will be formed by the screen since the reflection of the object and its refracted image will both occur. The screen here acts as a plane mirror.