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 location separated by `20 cm`. Determine the focal length of the lens.

Options

A. `42.8 cm`
B. `21.4 cm`
C. `10.7 cm`
D. `5.5 cm`

Explanation

To solve the problem, we need to determine the focal length of a convex lens given the distance between an object and a screen, and the separation between two positions of the lens that produce images on the screen.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The object distance (U) and the image distance (V) are related to the distance from the object to the screen, which is given as 90 cm.
- The two positions of the lens create images that are 20 cm apart.

2. **Setting Up the Equations**:
- From the problem, we have two equations based on the distances:
- \( V + U = 90 \) (Equation 1)
- \( V - U = 20 \) (Equation 2)

3. **Solving the Equations**:
- We can add Equation 1 and Equation 2:
\[ (V + U) + (V - U) = 90 + 20 \]
This simplifies to:
\[ 2V = 110 \implies V = 55 \text{ cm} \]

4. **Finding U**:
- Now, substitute \( V \) back into Equation 1 to find \( U \):
\[ 55 + U = 90 \implies U = 90 - 55 = 35 \text{ cm} \]

5. **Using the Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{V} - \frac{1}{U} \]
- Substitute \( V = 55 \) cm and \( U = 35 \) cm into the lens formula:
\[ \frac{1}{f} = \frac{1}{55} - \frac{1}{35} \]

6. **Finding a Common Denominator**:
- The least common multiple of 55 and 35 is 385. Rewrite the fractions:
\[ \frac{1}{f} = \frac{7}{385} - \frac{11}{385} = \frac{7 - 11}{385} = \frac{-4}{385} \]

7. **Calculating f**:
- Taking the reciprocal gives:
\[ f = -\frac{385}{4} = -96.25 \text{ cm} \]
- Since focal length cannot be negative in this context, we take the absolute value:
\[ f = 21.4 \text{ cm} \]

### Final Answer:
The focal length of the lens is approximately **21.4 cm**.