Two similar thin equi-convex lenses, of focal f each, are kept coaxially in contact with each other such that the focal length of the combination is `F_(1)`, When the space between the two lens is filled with glycerin (which has the same refractive index `(mu=1.5)` as that of glass) then the equivlent focal length is `F_(2)`, The ratio `F_(1): F_(2)` will be

Options

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

Explanation

To solve the problem, we need to find the ratio of the focal lengths \( F_1 \) and \( F_2 \) of two similar thin equi-convex lenses in two different configurations.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two identical equi-convex lenses, each with a focal length \( f \).
- When they are placed in contact with each other, we need to find the focal length \( F_1 \) of the combination.

2. **Finding \( F_1 \)**:
- For two thin lenses in contact, the formula for the equivalent focal length \( F \) is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
- Since both lenses are identical, we can write:
\[ \frac{1}{F_1} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \]
- Therefore, the focal length \( F_1 \) is:
\[ F_1 = \frac{f}{2} \]

3. **Finding \( F_2 \)**:
- Now, when the space between the two lenses is filled with glycerin (which has the same refractive index as glass, \( \mu = 1.5 \)), we need to find the new focal length \( F_2 \).
- The effective focal length of the combination can be calculated as:
\[ \frac{1}{F_2} = \frac{1}{f} + \frac{1}{f} - \frac{1}{f} \]
- The term \( -\frac{1}{f} \) comes from the fact that the space filled with glycerin acts like a concave lens. Thus:
\[ \frac{1}{F_2} = \frac{2}{f} - \frac{1}{f} = \frac{1}{f} \]
- Therefore, the focal length \( F_2 \) is:
\[ F_2 = f \]

4. **Finding the Ratio \( F_1 : F_2 \)**:
- Now we can find the ratio of \( F_1 \) to \( F_2 \):
\[ \frac{F_1}{F_2} = \frac{\frac{f}{2}}{f} = \frac{1}{2} \]
- Thus, the ratio \( F_1 : F_2 \) is:
\[ F_1 : F_2 = 1 : 2 \]

### Final Answer:
The ratio \( F_1 : F_2 \) is \( 1 : 2 \).