The focal length of a mirror is give by `(1)/(f)=(1)/(u)+(1)/(v)`, where 'u' represents object distance, 'v' image distance and 'f' local length. The maximum relative error is

Options

A. `(Delta f)/(f)=(Delta u)/(u)+(Delta v)/(v)`
B. `-(df)/(f^(2))=(-(du)/(u^(2)))+(-(dv)/(v^(2)))`
C. `(Delta f)/(f)=(Delta u)/(u)+(Delta v)/(v)+(Delta(u+v))/(u+v)`
D. `(Delta f)/(f)=(Delta u)/(u)+(Delta v)/(v)+(Delta u)/(u+v)+(Delta v)/(u+v)`

Answer

To solve the problem of finding the maximum relative error in the focal length of a mirror given by the formula \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \), we can follow these steps:

### Step 1: Understand the relationship
The formula for the focal length \( f \) can be rearranged as:
\[ f = \frac{uv}{u + v} \]
where \( u \) is the object distance and \( v \) is the image distance.

### Step 2: Identify the relative errors
The maximum relative error in \( f \) can be expressed in terms of the relative errors in \( u \) and \( v \). The formula for the maximum relative error in a quotient or product is the sum of the relative errors of the individual quantities involved.

### Step 3: Write the expression for relative error
The relative error in \( f \) can be given by:
\[ \frac{\Delta f}{f} = \frac{\Delta u}{u} + \frac{\Delta v}{v} + \frac{\Delta (u + v)}{(u + v)} \]
where \( \Delta u \) and \( \Delta v \) are the absolute errors in \( u \) and \( v \) respectively.

### Step 4: Simplify the expression
The term \( \Delta (u + v) \) can be expressed as:
\[ \Delta (u + v) = \Delta u + \Delta v \]
Thus, we can rewrite the relative error as:
\[ \frac{\Delta f}{f} = \frac{\Delta u}{u} + \frac{\Delta v}{v} + \frac{\Delta u + \Delta v}{u + v} \]

### Step 5: Final expression
The final expression for the maximum relative error in \( f \) becomes:
\[ \frac{\Delta f}{f} = \frac{\Delta u}{u} + \frac{\Delta v}{v} + \frac{\Delta u}{u + v} + \frac{\Delta v}{u + v} \]

### Conclusion
From the options provided, we can see that the correct answer matches with options C and D, which include the terms we derived.