In a concave mirror experiment, an object is placed at a distance `x_(1)` from the focus and the image is formed at a distance `x_(2)` from the focus. The focal length of the mirror would be

Options

A. `x_(1)x_(2)`
B. `sqrt(x_(1)x_(2))`
C. `(x_(1)+x_(2))/(2)`
D. `sqrt((x_(1))/(x_(2)))`

Explanation

To find the focal length of a concave mirror when the object is placed at a distance \( x_1 \) from the focus and the image is formed at a distance \( x_2 \) from the focus, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Object and Image Distances:**
- Let the focal length of the mirror be \( f \).
- The object distance \( u \) from the mirror is given by:
\[ u = f + x_1 \]
- The image distance \( v \) from the mirror is given by:
\[ v = f + x_2 \]

2. **Use the Mirror Formula:**
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
- Substitute the values of \( u \) and \( v \):
\[ \frac{1}{f} = \frac{1}{(f + x_1)} + \frac{1}{(f + x_2)} \]

3. **Combine the Right Side:**
- To combine the fractions on the right side, we find a common denominator:
\[ \frac{1}{f} = \frac{(f + x_2) + (f + x_1)}{(f + x_1)(f + x_2)} \]
- Simplifying the numerator:
\[ \frac{1}{f} = \frac{2f + x_1 + x_2}{(f + x_1)(f + x_2)} \]

4. **Cross Multiply:**
- Cross multiplying gives:
\[ (f + x_1)(f + x_2) = f(2f + x_1 + x_2) \]

5. **Expand Both Sides:**
- Expanding the left side:
\[ f^2 + f x_2 + f x_1 + x_1 x_2 = 2f^2 + f x_1 + f x_2 \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ f^2 + x_1 x_2 = 0 \]
- This simplifies to:
\[ f^2 = x_1 x_2 \]

7. **Finding the Focal Length:**
- Taking the square root gives:
\[ f = \sqrt{x_1 x_2} \]

### Final Answer:
The focal length of the concave mirror is:
\[ f = \sqrt{x_1 x_2} \]