With a concave mirror, an object is placed at a distance `x_(1)` from the principal focus, on the principal axis. The image is formed at a distance `x_(2)` from the principal focus. The focal length of the mirror is

Options

A. `x_(1)x_(2)`
B. `(x_(1)+x_(2))/(2)`
C. `sqrt((x_(1))/(x_(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 principal focus and the image is formed at a distance \( x_2 \) from the principal focus, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the 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 relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

3. **Substitute the values of \( u \) and \( v \)**:
Substituting the expressions for \( u \) and \( v \) into the mirror formula gives:
\[ \frac{1}{f} = \frac{1}{f + x_1} + \frac{1}{f + x_2} \]

4. **Find a common denominator**:
The common denominator for the right-hand side is \( (f + x_1)(f + x_2) \):
\[ \frac{1}{f} = \frac{(f + x_2) + (f + x_1)}{(f + x_1)(f + x_2)} \]
Simplifying the numerator:
\[ \frac{1}{f} = \frac{2f + x_1 + x_2}{(f + x_1)(f + x_2)} \]

5. **Cross-multiply to eliminate the fractions**:
Cross-multiplying gives:
\[ (f + x_1)(f + x_2) = f(2f + x_1 + x_2) \]

6. **Expand both sides**:
Expanding the left-hand side:
\[ f^2 + (x_1 + x_2)f + x_1x_2 = 2f^2 + f(x_1 + x_2) \]

7. **Rearranging the equation**:
Rearranging the equation results in:
\[ f^2 + (x_1 + x_2)f + x_1x_2 - 2f^2 - f(x_1 + x_2) = 0 \]
Simplifying gives:
\[ -f^2 + x_1x_2 = 0 \]
or
\[ f^2 = x_1x_2 \]

8. **Taking the square root**:
Finally, taking the square root of both sides yields:
\[ f = \sqrt{x_1 x_2} \]

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