A man can see clearly up to 3 metres. Prescribes a lens for his spectacles so that he can see clearly up to `12 ` metres

Options

A. `-3//4D`
B. `3D`
C. `-1//4D`
D. `-4D`

Explanation

To solve the problem of prescribing a lens for a man who can see clearly up to 3 meters but needs to see clearly up to 12 meters, we will use the lens formula and the concept of power of the lens.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The nearest point of clear vision for the man (his near point) is \( V = -3 \) meters (the negative sign indicates that it is a virtual image).
- The farthest point he wants to see clearly (the object distance) is \( U = -12 \) meters (again, the negative sign indicates that it is on the same side as the object).

2. **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, \( V \) is the image distance, and \( U \) is the object distance.

3. **Substitute the Values:**
Substitute \( V = -3 \) m and \( U = -12 \) m into the lens formula:
\[ \frac{1}{F} = \frac{1}{-3} - \frac{1}{-12} \]

4. **Calculate the Right Side:**
To perform the subtraction, find a common denominator:
\[ \frac{1}{F} = -\frac{1}{3} + \frac{1}{12} \]
The common denominator of 3 and 12 is 12:
\[ \frac{1}{F} = -\frac{4}{12} + \frac{1}{12} = -\frac{3}{12} = -\frac{1}{4} \]

5. **Find the Focal Length:**
From the equation:
\[ \frac{1}{F} = -\frac{1}{4} \]
Therefore, the focal length \( F \) is:
\[ F = -4 \text{ meters} \]

6. **Calculate the Power of the Lens:**
The power \( P \) of a lens is given by:
\[ P = \frac{1}{F} \text{ (in meters)} \]
Thus:
\[ P = \frac{1}{-4} = -0.25 \text{ diopters} \]

7. **Conclusion:**
The man needs a lens with a power of \(-0.25\) diopters. This indicates that he requires a concave lens.