Identical objective lenses are used to make Astronomical telescope and Galilean telescope. Focal length of objective for both is `f_(o)`. Focal length of eyepiece for both is also same and equal to `f_(e)`. Final image is formed at infinity in both the telescopes.

Options

A. Magnification for both the telescopes will be same.
B. Tube lengths of the telescopes differ by `2f_(e)`.
C. Length of Astronomical telescope is less than Galilean.
D. Length of Astronomical telescope is more than Galilean.

Explanation

To solve the problem regarding the Astronomical and Galilean telescopes using identical objective lenses, we will follow these steps:

### Step 1: Understand the Components of Each Telescope
- **Astronomical Telescope**: It consists of two positive lenses (objective and eyepiece). The final image is formed at infinity.
- **Galilean Telescope**: It consists of a positive objective lens and a negative eyepiece lens. The final image is also formed at infinity.

### Step 2: Determine the Tube Lengths
- **For the Astronomical Telescope**: The tube length (L_a) is given by the sum of the focal lengths of the objective and eyepiece:
\[ L_a = |f_o| + |f_e| \]

- **For the Galilean Telescope**: The tube length (L_g) is given by the difference of the focal lengths of the objective and eyepiece:
\[ L_g = |f_o| - |f_e| \]

### Step 3: Calculate the Difference in Tube Lengths
To find the difference in tube lengths, we subtract the tube length of the Galilean telescope from that of the Astronomical telescope:
\[ \Delta L = L_a - L_g = (|f_o| + |f_e|) - (|f_o| - |f_e|) = 2|f_e| \]

### Step 4: Analyze the Magnification
The magnification (M) for both telescopes can be calculated as follows:
- **For the Astronomical Telescope**:
\[ M_a = -\frac{f_o}{f_e} \]

- **For the Galilean Telescope**:
\[ M_g = -\frac{f_o}{f_e} \]

Since both magnifications are equal, we can conclude that the magnification for both telescopes is the same.

### Step 5: Compare the Lengths of the Telescopes
From the calculations:
- The tube length of the Astronomical telescope is greater than that of the Galilean telescope, as:
\[ L_a > L_g \]

Thus, the statement that the length of the Astronomical telescope is less than the Galilean telescope is incorrect.

### Final Summary
1. The tube lengths differ by \(2|f_e|\).
2. The magnification for both telescopes is the same.
3. The length of the Astronomical telescope is greater than that of the Galilean telescope.

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