A short sighted person can see objects most distinctly at a distance of 16 cm. If he wears spectacles at a distance of 1 cm from the eye, then their focal length to see distinctly at a distance of 26 cm

Options

A. 25 cm, convex
B. 25 cm, concave
C. 37.5 cm, convex
D. 37.5 cm, concave

Explanation

To solve the problem, we need to determine the focal length of the spectacles needed for a short-sighted person who can see objects most distinctly at a distance of 16 cm. The person wants to see objects at a distance of 26 cm while wearing spectacles that are 1 cm away from the eye.

### Step-by-Step Solution:

1. **Identify the Distances:**
- The maximum distance the short-sighted person can see clearly without spectacles is \( d_{max} = 16 \, \text{cm} \).
- The distance at which the person wants to see clearly with spectacles is \( d_{target} = 26 \, \text{cm} \).
- The distance from the spectacles to the eye is \( d_{lens-eye} = 1 \, \text{cm} \).

2. **Calculate the Object Distance (U):**
- The object distance from the lens (U) can be calculated as:
\[ U = d_{target} - d_{lens-eye} = 26 \, \text{cm} - 1 \, \text{cm} = 25 \, \text{cm} \]
- Since the object is in front of the lens, we take the object distance as negative:
\[ U = -25 \, \text{cm} \]

3. **Calculate the Image Distance (V):**
- The image distance (V) is the distance at which the short-sighted person can see clearly:
\[ V = d_{max} - d_{lens-eye} = 16 \, \text{cm} - 1 \, \text{cm} = 15 \, \text{cm} \]
- Again, since the image is formed on the same side as the object, we take it as negative:
\[ V = -15 \, \text{cm} \]

4. **Use the Lens Formula:**
- The lens formula is given by:
\[ \frac{1}{F} = \frac{1}{V} - \frac{1}{U} \]
- Substituting the values of V and U:
\[ \frac{1}{F} = \frac{1}{-15} - \frac{1}{-25} \]

5. **Calculate the Right Side:**
- Finding a common denominator (LCM of 15 and 25 is 75):
\[ \frac{1}{F} = -\frac{5}{75} + \frac{3}{75} = -\frac{2}{75} \]

6. **Calculate the Focal Length (F):**
- Taking the reciprocal gives:
\[ F = -\frac{75}{2} = -37.5 \, \text{cm} \]
- The negative sign indicates that the lens is a concave lens.

### Final Answer:
The focal length of the spectacles required for the person to see distinctly at a distance of 26 cm is \( 37.5 \, \text{cm} \) (concave lens).

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