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

Options

A. 18 cm
B. 21.4 cm
C. 60 cm
D. 85.6 cm

Answer

To solve the problem, we need to determine the focal length of a convex lens given the distance between an object and a screen, along with the information about the two positions of the lens.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The distance from the object to the screen is given as 90 cm.
- The two positions of the lens create images on the screen that are 20 cm apart.

2. **Setting Up the Equations**:
- Let \( u_1 \) be the object distance for the first position of the lens and \( v_1 \) be the image distance for the first position.
- According to the problem, we have:
\[ u_1 + v_1 = 90 \quad \text{(1)} \]
- For the second position of the lens, let \( u_2 \) and \( v_2 \) be the object and image distances, respectively. Since the lens is moved 20 cm, we can express this as:
\[ v_1 - u_1 = 20 \quad \text{(2)} \]

3. **Solving the Equations**:
- From equation (1), we can express \( v_1 \) in terms of \( u_1 \):
\[ v_1 = 90 - u_1 \quad \text{(3)} \]
- Substitute equation (3) into equation (2):
\[ (90 - u_1) - u_1 = 20 \]
\[ 90 - 2u_1 = 20 \]
\[ 2u_1 = 70 \]
\[ u_1 = 35 \, \text{cm} \]
- Now substitute \( u_1 \) back into equation (3) to find \( v_1 \):
\[ v_1 = 90 - 35 = 55 \, \text{cm} \]

4. **Using the Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Substitute \( v_1 \) and \( u_1 \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{55} - \frac{1}{-35} \]
\[ \frac{1}{f} = \frac{1}{55} + \frac{1}{35} \]
- To combine these fractions, find a common denominator (which is 385):
\[ \frac{1}{f} = \frac{7}{385} + \frac{11}{385} = \frac{18}{385} \]
- Therefore, the focal length \( f \) is:
\[ f = \frac{385}{18} \approx 21.4 \, \text{cm} \]

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