When an object is at .distances of `mu_(1)` and `mu_(2)` from the poles of a concave mirror, images of the same size are formed. The - focal length, of the mirror is

Options

A. `|mu_(1)+mu_(2)`|
B. |`mu_(2)+mu_(1)`|
C. |`(mu_(1)+mu_(2))/2`|
D. |`(mu_(1)-mu_(2))/2`|

Answer

To solve the problem, we will use the mirror formula and the concept of magnification. Let's break it down step by step.

### Step 1: Understand the given information
We have two object distances, \( u_1 \) and \( u_2 \), from the pole of a concave mirror, and we know that the images formed at these distances are of the same size. This means that the magnification for both positions is equal to 1.

### Step 2: Use 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.

### Step 3: Write the magnification formula
The magnification \( M \) for a mirror is given by:
\[ M = -\frac{v}{u} \]
Since the images are of the same size, we can set the magnification for both distances equal to 1:
\[ M_1 = -\frac{v_1}{u_1} = 1 \quad \text{and} \quad M_2 = -\frac{v_2}{u_2} = 1 \]
This implies that:
\[ v_1 = -u_1 \quad \text{and} \quad v_2 = -u_2 \]

### Step 4: Substitute the image distances into the mirror formula
For the first object distance \( u_1 \):
\[ \frac{1}{f} = \frac{1}{-u_1} + \frac{1}{-u_1} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{2}{u_1} \]
Thus, we can express \( f \) as:
\[ f = -\frac{u_1}{2} \]

For the second object distance \( u_2 \):
\[ \frac{1}{f} = \frac{1}{-u_2} + \frac{1}{-u_2} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{2}{u_2} \]
Thus, we can express \( f \) as:
\[ f = -\frac{u_2}{2} \]

### Step 5: Set the two expressions for focal length equal
Since both expressions for \( f \) must be equal:
\[ -\frac{u_1}{2} = -\frac{u_2}{2} \]
This leads to:
\[ u_1 = u_2 \]

### Step 6: Find the focal length in terms of \( u_1 \) and \( u_2 \)
From the earlier steps, we can find the average focal length:
\[ f = \frac{u_1 + u_2}{2} \]

### Conclusion
Thus, the focal length \( f \) of the concave mirror is given by:
\[ f = \frac{u_1 + u_2}{2} \]