A screen is placed away from an object. The image of the object on the screen is $90\;{\text{cm}}$formed by a convex lens at two different locations separated by $20\;{\text{cm}}$. Find the focal length of the lens.
(A) $42.8\;{\text{cm}}$
(B) $21.4\;{\text{cm}}$
(C) $10.7\;{\text{cm}}$
(D) $5.5\;{\text{cm}}$

Hint: Here, by using the formula, we can simply find out the lens's focal length. The distance between the screen and the object is given to us as well as the distance between two locations of the images by which we can simply find out the lens's focal length.
Formula Used: We will use the following formula to find out the result
$f = \dfrac{{{D^2} - {d^2}}}{{4D}}$
Where
$f$ is the focal length of the lens
${\text{D}}$ is the distance between the object and the screen
${\text{d}}$ is the distance between the two locations of the convex lens

Complete Step-by-Step Solution:
The following information is provided to us in the question
The distance between the screen and the object ${\text{D}} = 90\;{\text{cm}}$
The distance between the two locations of the convex lens ${\text{d}} = 20\;{\text{cm}}$
Now, let us use our formula provided above
$f = \dfrac{{{D^2} - {d^2}}}{{4D}}$
Upon substituting known values, we get
$f = \dfrac{{{{90}^2} - {{20}^2}}}{{4 \times 90}}$
$ \Rightarrow f = \dfrac{{8100 - 400}}{{360}}$
Upon simplifying, we get
$\therefore f = 21.4\;{\text{cm}}$

Hence, the correct option is (B.)

Additional Information: A lens is a transmissive optical device that uses refraction to focus or disperse a light beam. A simple lens is made up of a single piece of transparent material, while a compound lens is made up of several simple lenses (elements), usually arranged along a common axis.

Note: The proper definition of focal length should be known to us. It states that the distance from the centre of a lens to the point at which its image is focused is measured by focal length