In an astronomincal telescope, the distance between the objective and the eyepiece is 36cm and the final image is formed at infinity. The focal length `f_(0)` of the objective and the focall length `f_(e)` of th eeyepiece are

Options

A. `f_(0)=45` cm and `f_(e)=-9cm`
B. `f_(0)=50` cm and `f_(e)=10cm`
C. `f_(0)=7.2` cm and `f_(e)=5cm`
D. `f_(0)=30` cm and `f_(e)=6cm`

Explanation

To solve the problem regarding the focal lengths of the objective and eyepiece in an astronomical telescope, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
In an astronomical telescope, the distance between the objective lens and the eyepiece lens is given as \( L = 36 \, \text{cm} \). The final image is formed at infinity.

2. **Use the Lens Formula**:
The relationship between the focal lengths of the objective lens \( f_0 \) and the eyepiece lens \( f_e \) can be expressed as:
\[ L = f_0 + f_e \]
where \( L \) is the distance between the two lenses.

3. **Substitute the Given Value**:
Substitute \( L = 36 \, \text{cm} \) into the equation:
\[ f_0 + f_e = 36 \]

4. **Identify Possible Focal Lengths**:
We need to find pairs of \( f_0 \) and \( f_e \) that satisfy this equation. We can check the options provided in the question.

5. **Check Each Option**:
- **Option A**: \( f_0 = 45 \, \text{cm}, f_e = 9 \, \text{cm} \)
\[ 45 + 9 = 54 \quad (\text{Not valid}) \]
- **Option B**: \( f_0 = 50 \, \text{cm}, f_e = 10 \, \text{cm} \)
\[ 50 + 10 = 60 \quad (\text{Not valid}) \]
- **Option C**: \( f_0 = 7.2 \, \text{cm}, f_e = 5 \, \text{cm} \)
\[ 7.2 + 5 = 12.2 \quad (\text{Not valid}) \]
- **Option D**: \( f_0 = 30 \, \text{cm}, f_e = 6 \, \text{cm} \)
\[ 30 + 6 = 36 \quad (\text{Valid}) \]

6. **Conclusion**:
The valid pairs of focal lengths that satisfy the equation \( f_0 + f_e = 36 \, \text{cm} \) are:
- \( f_0 = 30 \, \text{cm} \) and \( f_e = 6 \, \text{cm} \)

### Final Answer:
- The focal lengths are \( f_0 = 30 \, \text{cm} \) and \( f_e = 6 \, \text{cm} \).