The focal length of the objective and eye piece of a telescope are respectively `100 cm and 2 cm`. The moon subtends and angle of `0.5^@`, the angle subtended by the moon's image will be.

Options

A. `100^(@)`
B. `50^(@)`
C. `25^(@)`
D. `10^(@)`

Answer

To solve the problem, we need to find the angle subtended by the moon's image (β) using the relationship between the angles subtended by the object (α) and the focal lengths of the telescope's objective (f₀) and eyepiece (fₑ).

### Step-by-step Solution:

1. **Identify the given values**:
- Focal length of the objective (f₀) = 100 cm
- Focal length of the eyepiece (fₑ) = 2 cm
- Angle subtended by the moon (α) = 0.5 degrees

2. **Use the formula for the telescope**:
The relationship between the angles is given by the formula:
\[ \frac{\beta}{\alpha} = \frac{f₀}{fₑ} \]
where:
- β = angle subtended by the image
- α = angle subtended by the object

3. **Substitute the known values into the formula**:
\[ \frac{\beta}{0.5} = \frac{100}{2} \]

4. **Calculate the ratio**:
\[ \frac{100}{2} = 50 \]
So, we have:
\[ \frac{\beta}{0.5} = 50 \]

5. **Solve for β**:
Multiply both sides by 0.5:
\[ \beta = 50 \times 0.5 \]
\[ \beta = 25 \text{ degrees} \]

6. **Conclusion**:
The angle subtended by the moon's image (β) is 25 degrees.

### Final Answer:
The angle subtended by the moon's image is **25 degrees**.

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