An object is placed in front of a spherical mirror of focal length f. If x and x' recpectively represent the distance of the object and the image from the focus, then

Options

A. `f=x+x'`
B. `f^(2)=x x`
C. `f=|x-x'|`
D. `f=x x`

Explanation

To derive the relationship between the distances of the object and the image from the focus of a spherical mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mirror Formula**:
The mirror formula is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
where \( v \) is the image distance, \( u \) is the object distance, and \( f \) is the focal length of the mirror.

2. **Define Distances from the Focus**:
Let \( x \) be the distance of the object from the focus, and \( x' \) be the distance of the image from the focus. Thus, we can express:
\[ u = x + f \quad \text{(since object distance is measured from the mirror)} \]
\[ v = x' + f \quad \text{(since image distance is also measured from the mirror)} \]

3. **Substitute into the Mirror Formula**:
Substitute \( u \) and \( v \) into the mirror formula:
\[ \frac{1}{x' + f} + \frac{1}{x + f} = \frac{1}{f} \]

4. **Multiply through by \( (x' + f)(x + f)f \)**:
To eliminate the fractions, multiply through by \( (x' + f)(x + f)f \):
\[ f(x + f) + f(x' + f) = (x' + f)(x + f) \]

5. **Expand the Equation**:
Expanding both sides gives:
\[ fx + f^2 + fx' + f^2 = x'x + fx + fx' + f^2 \]

6. **Simplify the Equation**:
Cancel out the common terms on both sides:
\[ f^2 = x'x - fx - fx' \]

7. **Rearranging the Terms**:
Rearranging gives:
\[ x \cdot x' = f^2 + fx + fx' \]

8. **Final Relation**:
Thus, we arrive at the relationship:
\[ x \cdot x' = f^2 \]

### Conclusion:
The derived relationship between the distances of the object and the image from the focus of a spherical mirror is:
\[ x \cdot x' = f^2 \]