The relation between u, v and f for a mirror given by:

Options

A. `f=(uv)/(u-v)`
B. `f=(2uxxv)/(u+v)`
C. `f=(uxxv)/(u+v)`
D. none of these

Answer

To solve the question regarding the relation between u, v, and f for a mirror, we will follow these steps:

### Step 1: Write down the mirror formula
The mirror formula relates the object distance (u), the image distance (v), and the focal length (f) of a mirror. The formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

### Step 2: Rearrange the formula to find f
To express f in terms of u and v, we need to manipulate the equation. We start by finding a common denominator for the right side of the equation:
\[ \frac{1}{f} = \frac{u + v}{uv} \]

### Step 3: Take the reciprocal to solve for f
Taking the reciprocal of both sides gives us:
\[ f = \frac{uv}{u + v} \]

### Step 4: Analyze the options
Now we compare our derived formula with the given options. We have:
- \( f = \frac{uv}{u + v} \)

### Step 5: Identify the correct option
Since our derived formula matches one of the options exactly, we can conclude that this is the correct answer. The other options either have incorrect signs or additional factors that do not match our derived formula.

### Final Answer
The correct relation between u, v, and f for a mirror is:
\[ f = \frac{uv}{u + v} \]

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