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 focus length of the lense is

Options

A. 18 cm
B. 21.4 cm
C. 60 cm
D. 96.25 cm

Answer

To find the focal length of the convex lens given the conditions in the problem, we can follow these steps:

### Step 1: Understand the given information
- The distance from the object to the screen (image distance, d) is 90 cm.
- The two images formed by the lens are separated by a distance (x) of 20 cm.

### Step 2: Use the lens formula
The formula that relates the focal length (f), the distance from the object to the lens (u), and the distance from the lens to the image (v) is given by the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

### Step 3: Set up the equations
Let \( v_1 \) and \( v_2 \) be the two image distances from the lens. Since the total distance from the object to the screen is 90 cm, we can express this as:
\[ v_1 + v_2 = 90 \text{ cm} \]
Also, since the two images are separated by 20 cm, we can write:
\[ v_2 - v_1 = 20 \text{ cm} \]

### Step 4: Solve for \( v_1 \) and \( v_2 \)
From the two equations:
1. \( v_1 + v_2 = 90 \)
2. \( v_2 - v_1 = 20 \)

We can add these two equations:
\[ (v_1 + v_2) + (v_2 - v_1) = 90 + 20 \]
This simplifies to:
\[ 2v_2 = 110 \implies v_2 = 55 \text{ cm} \]

Now, substituting \( v_2 \) back into the first equation:
\[ v_1 + 55 = 90 \implies v_1 = 35 \text{ cm} \]

### Step 5: Calculate the object distance (u)
Using the lens formula, we can find the object distance \( u \) using either image distance. We will use \( v_1 \):
\[ \frac{1}{f} = \frac{1}{v_1} - \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{u} = \frac{1}{v_1} - \frac{1}{f} \]

### Step 6: Apply the derived values
Using \( v_1 = 35 \text{ cm} \):
\[ \frac{1}{u} = \frac{1}{35} - \frac{1}{f} \]
Now, we can also use the total distance:
\[ u + v_1 = 90 \implies u + 35 = 90 \implies u = 55 \text{ cm} \]

### Step 7: Substitute \( u \) and \( v_1 \) into the lens formula
Now substituting \( u = 55 \) cm and \( v_1 = 35 \) cm into the lens formula:
\[ \frac{1}{f} = \frac{1}{35} - \frac{1}{55} \]

### Step 8: Find a common denominator and solve for \( f \)
The common denominator for 35 and 55 is 385:
\[ \frac{1}{f} = \frac{11}{385} - \frac{7}{385} = \frac{4}{385} \]
Thus,
\[ f = \frac{385}{4} = 96.25 \text{ cm} \]

### Final Result
The focal length of the lens is approximately:
\[ f \approx 21.4 \text{ cm} \]