A screen is placed 90 𝑐𝑚 from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by $20\text{cm}$. Determine the focal length of the lens.

Step 1: Draw the diagram.

Step 2: Find the values of $u_1$ and $v_1$.
The image of the object can be located on the screen for two-position of convex lens such that u and v are exchanged.
The separation between two positions of the lens is,
$d=20\text{cm}$
From the figure,
$u_1+v_1=90\text{cm}$
$v_1-u_1=20\text{cm}$
Solving,
$v_1=55\text{cm},u_1=35\text{cm}$.

Step 3: Find the value of focal length of lens.
Formula Used: $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
From lens formula, by putting the values of $u_1\text{and}{v}_1$,
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\frac{1}{55}-\frac{1}{-35}=\frac{1}{f}$
$\frac{1}{f}=\frac{1}{55}+\frac{1}{35}$
$\frac{1}{f}=\frac{55+35}{1925}$
$f=21.4\text{cm}$.
Final Answer : $f=21.4\text{cm}$.