Opera glass have a minimum length of 20 cm and a magnifying power of 5 when viewing distant objects. The focal lengths of lenses used are

Options

A. 25 cm, 5 cm
B. 25 cm, -5 cm
C. `(10/5)cm,(50/3)cm`
D. 15 cm, -10cm

Explanation

To find the focal lengths of the lenses used in the opera glass, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Minimum length of the opera glass, \( L = 20 \, \text{cm} \)
- Magnifying power (magnification), \( M = 5 \)

### Step 2: Relate Length and Focal Lengths
In a telescope (or opera glass), the minimum length \( L \) can be expressed as:
\[ L = f_o - f_e \]
where:
- \( f_o \) is the focal length of the objective lens
- \( f_e \) is the focal length of the eyepiece lens

### Step 3: Use the Magnification Formula
The magnification \( M \) for a telescope is given by:
\[ M = \frac{f_o}{f_e} \]
Given that \( M = 5 \), we can express \( f_o \) in terms of \( f_e \):
\[ f_o = 5 f_e \]

### Step 4: Substitute into the Length Equation
Now, we substitute \( f_o \) from the magnification equation into the length equation:
\[ 20 = (5 f_e) - f_e \]
This simplifies to:
\[ 20 = 4 f_e \]

### Step 5: Solve for \( f_e \)
To find \( f_e \):
\[ f_e = \frac{20}{4} = 5 \, \text{cm} \]

### Step 6: Find \( f_o \)
Now we can find \( f_o \) using the relationship we established:
\[ f_o = 5 f_e = 5 \times 5 = 25 \, \text{cm} \]

### Final Result
The focal lengths of the lenses used in the opera glass are:
- Focal length of the eyepiece, \( f_e = 5 \, \text{cm} \)
- Focal length of the objective, \( f_o = 25 \, \text{cm} \)