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

Options

A. (x + y)
B. (x - y)
C. `(xy)/(x+y)`
D. `(xy)/(y-x)`

Explanation

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 \( f_1 = x \) (positive value).
- Let the focal length of the concave lens be \( f_2 = y \) (negative value).

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

3. **Substitute the Focal Lengths**:
- Substitute \( f_1 \) and \( f_2 \) into the formula:
\[ \frac{1}{f} = \frac{1}{x} + \frac{1}{y} \]

4. **Convert the Concave Lens Focal Length**:
- Since the focal length of the concave lens is negative, we can write:
\[ \frac{1}{f} = \frac{1}{x} - \frac{1}{y} \]

5. **Combine the Fractions**:
- To combine the fractions, find a common denominator:
\[ \frac{1}{f} = \frac{y - x}{xy} \]

6. **Find the Focal Length of the Combination**:
- Taking the reciprocal gives us the focal length of the combination:
\[ f = \frac{xy}{y - x} \]

### Final Answer:
The focal length of the combination of the convex lens and concave lens placed in contact is:
\[ f = \frac{xy}{y - x} \]