In a concave mirror, an object is placed at a distance `d_1` from the focus and the real image is formed aat a distance `d_2` from the focus. Then the focal length of the mirror is :

Options

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

Explanation

To solve the problem, we will use the mirror formula and the given distances from the focus of the concave mirror.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a concave mirror with an object placed at a distance \(d_1\) from the focus.
- A real image is formed at a distance \(d_2\) from the focus.
- We need to find the focal length \(F\) of the mirror.

2. **Define Object and Image Distances**:
- The object distance \(u\) from the mirror's pole is given by:
\[ u = F + d_1 \]
- The image distance \(v\) from the mirror's pole is given by:
\[ v = F + d_2 \]

3. **Apply the Mirror Formula**:
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting the expressions for \(u\) and \(v\):
\[ \frac{1}{F} = \frac{1}{F + d_2} + \frac{1}{F + d_1} \]

4. **Combine the Fractions**:
- To combine the fractions on the right side, find a common denominator:
\[ \frac{1}{F} = \frac{(F + d_1) + (F + d_2)}{(F + d_1)(F + d_2)} \]
- This simplifies to:
\[ \frac{1}{F} = \frac{2F + d_1 + d_2}{(F + d_1)(F + d_2)} \]

5. **Cross-Multiply to Eliminate the Fractions**:
- Cross-multiplying gives:
\[ (F + d_1)(F + d_2) = F(2F + d_1 + d_2) \]

6. **Expand Both Sides**:
- Expanding the left side:
\[ F^2 + Fd_1 + Fd_2 + d_1d_2 = 2F^2 + Fd_1 + Fd_2 \]

7. **Rearranging the Equation**:
- Cancel out \(Fd_1\) and \(Fd_2\) from both sides:
\[ F^2 + d_1d_2 = 2F^2 \]
- Rearranging gives:
\[ d_1d_2 = 2F^2 - F^2 \]
\[ d_1d_2 = F^2 \]

8. **Solve for Focal Length**:
- Taking the square root of both sides:
\[ F = \sqrt{d_1d_2} \]

### Final Answer:
The focal length \(F\) of the mirror is:
\[ F = \sqrt{d_1 \cdot d_2} \]