With a concave mirrorr, 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 mirrorr 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 solve the problem, we need to find the focal length of a concave mirror given the distances of the object and the image from the principal focus. Let's break down the solution step by step.

### Step 1: Understand the problem
We have a concave mirror with:
- An object placed at a distance \( x_1 \) from the principal focus.
- An image formed at a distance \( x_2 \) from the principal focus.

### Step 2: Define the object and image distances
In the context of mirror formulas:
- The object distance \( u \) is given by:
\[ u = - (x_1 + f) \]
where \( f \) is the focal length of the mirror.

- The image distance \( v \) is given by:
\[ v = - (x_2 + f) \]

### Step 3: Apply the mirror formula
The mirror formula relates the object distance, image distance, and focal length:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the expressions for \( u \) and \( v \):
\[ \frac{1}{f} = \frac{1}{-(x_2 + f)} + \frac{1}{-(x_1 + f)} \]

### Step 4: Simplify the equation
Rearranging the equation gives:
\[ \frac{1}{f} = -\left(\frac{1}{x_2 + f} + \frac{1}{x_1 + f}\right) \]
This can be rewritten as:
\[ \frac{1}{f} = -\left(\frac{(x_1 + f) + (x_2 + f)}{(x_2 + f)(x_1 + f)}\right) \]
Thus:
\[ \frac{1}{f} = -\frac{x_1 + x_2 + 2f}{(x_2 + f)(x_1 + f)} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ (x_2 + f)(x_1 + f) = -f(x_1 + x_2 + 2f) \]

### Step 6: Expand both sides
Expanding both sides results in:
\[ x_1 x_2 + (x_1 + x_2)f + f^2 = -fx_1 - fx_2 - 2f^2 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ x_1 x_2 + (x_1 + x_2 + 2f)f = 0 \]

### Step 8: Solve for \( f \)
This can be simplified to:
\[ f^2 + (x_1 + x_2)f + x_1 x_2 = 0 \]
This is a quadratic equation in \( f \). Using the quadratic formula:
\[ f = \frac{-(x_1 + x_2) \pm \sqrt{(x_1 + x_2)^2 - 4x_1 x_2}}{2} \]

### Step 9: Find the focal length
Since we are looking for the focal length of a concave mirror, we take the negative root:
\[ f = \sqrt{x_1 x_2} \]

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