A person is not able to see objects farther than 80 cm clearly, while another person is not able to see objects beyond 120 cm, clearly. The powers of the lenses used by them for correct vision are in the ratio -

Options

A. `2:3`
B. `3:2`
C. `1:2`
D. `2:1`

Explanation

To solve the problem, we need to determine the powers of the lenses required for two individuals who have different near point distances. The first person can see objects clearly up to 80 cm, and the second person can see objects clearly up to 120 cm. We will find the ratio of the powers of the lenses needed for each person.

### Step 1: Understand the Concept of Power of a Lens
The power of a lens (P) is defined as the reciprocal of its focal length (f) in meters:
\[ P = \frac{1}{f} \]
The unit of power is diopters (D).

### Step 2: Identify the Focal Lengths
For the first person who cannot see beyond 80 cm, we need to determine the focal length of the lens required to correct their vision to see at infinity. The object distance (u) is -80 cm (negative because it is a real object for a lens), and we want the image distance (v) to be at infinity (v = ∞).

Using the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{f_1} = \frac{1}{\infty} - \frac{1}{-80} \]
\[ \frac{1}{f_1} = 0 + \frac{1}{80} \]
Thus,
\[ f_1 = 80 \text{ cm} = 0.8 \text{ m} \]

### Step 3: Calculate the Power for the First Person
Now, we can find the power of the lens for the first person:
\[ P_1 = \frac{1}{f_1} = \frac{1}{0.8} = 1.25 \text{ D} \]

### Step 4: Repeat for the Second Person
For the second person who cannot see beyond 120 cm, we will do the same:
The object distance (u) is -120 cm, and we want the image distance (v) to be at infinity.

Using the lens formula:
\[ \frac{1}{f_2} = \frac{1}{\infty} - \frac{1}{-120} \]
\[ \frac{1}{f_2} = 0 + \frac{1}{120} \]
Thus,
\[ f_2 = 120 \text{ cm} = 1.2 \text{ m} \]

### Step 5: Calculate the Power for the Second Person
Now, we can find the power of the lens for the second person:
\[ P_2 = \frac{1}{f_2} = \frac{1}{1.2} \approx 0.833 \text{ D} \]

### Step 6: Find the Ratio of the Powers
Now, we can find the ratio of the powers of the lenses used by both persons:
\[ \frac{P_1}{P_2} = \frac{1.25}{0.833} \]
To simplify this, we can multiply both sides by 1200 to eliminate the decimals:
\[ \frac{P_1}{P_2} = \frac{1250}{833} \approx \frac{3}{2} \]

### Final Answer
The ratio of the powers of the lenses used by the two persons is:
\[ \frac{P_1}{P_2} = \frac{3}{2} \]

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