Choose the correct relation between u, v and r for a spherical mirror:

Options

A. `R=(2uv)/(u+v)`
B. `R=2/(u+v)`
C. `R=(2(u+v))/"(uv)"`
D. none of these

Answer

To solve the question regarding the correct relation between \( u \), \( v \), and \( r \) for a spherical mirror, we will use the mirror formula and the relationship between the focal length and the radius of curvature.

### Step-by-Step Solution:

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

**Hint**: Remember that \( f \) is positive for concave mirrors and negative for convex mirrors.

2. **Relate Focal Length to Radius of Curvature**:
The focal length \( f \) is related to the radius of curvature \( r \) by the formula:
\[ f = \frac{r}{2} \]

**Hint**: This relationship holds true for all spherical mirrors.

3. **Substitute \( f \) in the Mirror Formula**:
Substitute \( f \) in the mirror formula:
\[ \frac{1}{\frac{r}{2}} = \frac{1}{v} + \frac{1}{u} \]
This simplifies to:
\[ \frac{2}{r} = \frac{1}{v} + \frac{1}{u} \]

**Hint**: When substituting, ensure to simplify correctly.

4. **Find a Common Denominator**:
To combine the fractions on the right side, find a common denominator:
\[ \frac{2}{r} = \frac{u + v}{uv} \]

**Hint**: The common denominator is the product of the two distances \( u \) and \( v \).

5. **Cross Multiply**:
Cross-multiplying gives:
\[ 2uv = r(u + v) \]

**Hint**: This step allows us to isolate \( r \).

6. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ r = \frac{2uv}{u + v} \]

**Hint**: This is the final relation you are looking for.

### Conclusion:
The correct relation between \( u \), \( v \), and \( r \) for a spherical mirror is:
\[ r = \frac{2uv}{u + v} \]