A screen is placed `80 cm` from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by `10 cm`. Calculate the focal length of the lens used.

Explanation

To solve the problem, we will use the displacement method for a convex lens. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Distance from the object to the screen (D) = 80 cm
- Distance between the two image locations (d) = 10 cm

### Step 2: Use the formula for focal length
According to the displacement method, the focal length (f) of the lens can be calculated using the formula:
\[ f = \frac{D^2 - d^2}{4D} \]

### Step 3: Substitute the values into the formula
Now, we will substitute the values of D and d into the formula:
\[ f = \frac{(80 \, \text{cm})^2 - (10 \, \text{cm})^2}{4 \times 80 \, \text{cm}} \]

### Step 4: Calculate \(D^2\) and \(d^2\)
Calculating \(D^2\) and \(d^2\):
\[ D^2 = 80^2 = 6400 \, \text{cm}^2 \]
\[ d^2 = 10^2 = 100 \, \text{cm}^2 \]

### Step 5: Substitute the squared values back into the formula
Now substituting these values into the formula:
\[ f = \frac{6400 \, \text{cm}^2 - 100 \, \text{cm}^2}{4 \times 80 \, \text{cm}} \]
\[ f = \frac{6300 \, \text{cm}^2}{320 \, \text{cm}} \]

### Step 6: Perform the division
Now, we will perform the division:
\[ f = \frac{6300}{320} \approx 19.6875 \, \text{cm} \]

### Step 7: Round off the answer
Rounding off to two decimal places, we get:
\[ f \approx 19.69 \, \text{cm} \]

### Final Answer
The focal length of the lens used is approximately \(19.69 \, \text{cm}\).

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