A person can see things clearly upto 3m what is the power of lens he should use so that he can see upto 12 m ?

Options

A. `+0.5 D`
B. `-0.5 D`
C. `+0.25D`
D. `-0.25D`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
The person can see clearly up to 3 meters (300 cm) and needs to see clearly up to 12 meters (1200 cm). We need to find the power of the lens required to achieve this.

### Step 2: Define the Variables
- The distance up to which the person can see clearly (u) = -300 cm (object distance is taken as negative in lens formula)
- The distance up to which the person wants to see clearly (v) = -1200 cm (image distance is also taken as negative)

### Step 3: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length of the lens.

### Step 4: Substitute the Values into the Lens Formula
Substituting the values of \( u \) and \( v \):
\[ \frac{1}{f} = \frac{1}{-1200} - \frac{1}{-300} \]

### Step 5: Calculate the Right Side
Calculating the right side:
\[ \frac{1}{f} = -\frac{1}{1200} + \frac{1}{300} \]
To combine these fractions, we need a common denominator. The least common multiple of 1200 and 300 is 1200.
\[ \frac{1}{300} = \frac{4}{1200} \]
So,
\[ \frac{1}{f} = -\frac{1}{1200} + \frac{4}{1200} = \frac{3}{1200} \]

### Step 6: Find the Focal Length
Now, we can find \( f \):
\[ f = \frac{1200}{3} = 400 \text{ cm} \]

### Step 7: Convert Focal Length to Meters
Convert the focal length from centimeters to meters:
\[ f = 400 \text{ cm} = 4 \text{ m} \]

### Step 8: Calculate the Power of the Lens
The power \( P \) of the lens is given by:
\[ P = \frac{1}{f} \text{ (in meters)} \]
So,
\[ P = \frac{1}{4} = 0.25 \text{ diopters} \]

### Step 9: Determine the Sign of Power
Since the lens is diverging (to correct for the person's vision), the power will be negative:
\[ P = -0.25 \text{ diopters} \]

### Final Answer
The power of the lens required is **-0.25 diopters**.

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