A convex lens of focal length A and a concave lens of focal length B are placed in contact. The focal length of the combination is

Options

A. (A+B)
B. (A-B)
C. `(AB)/((A+B))`
D. `(AB)/((B-A))`

Answer

To find the focal length of a combination of a convex lens and a concave lens placed in contact, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focal Lengths**:
- Let the focal length of the convex lens be \( f_1 = A \) (positive).
- Let the focal length of the concave lens be \( f_2 = -B \) (negative).

2. **Use the Lens Combination Formula**:
- The formula for the combined focal length \( f_c \) of two lenses in contact is given by:
\[ \frac{1}{f_c} = \frac{1}{f_1} + \frac{1}{f_2} \]

3. **Substitute the Values**:
- Substitute \( f_1 \) and \( f_2 \) into the formula:
\[ \frac{1}{f_c} = \frac{1}{A} + \frac{1}{-B} \]
- This simplifies to:
\[ \frac{1}{f_c} = \frac{1}{A} - \frac{1}{B} \]

4. **Combine the Fractions**:
- To combine the fractions, find a common denominator:
\[ \frac{1}{f_c} = \frac{B - A}{AB} \]

5. **Take the Reciprocal**:
- To find \( f_c \), take the reciprocal of both sides:
\[ f_c = \frac{AB}{B - A} \]

### Final Result:
The focal length of the combination of the convex lens and the concave lens is:
\[ f_c = \frac{AB}{B - A} \]