A convex mirror has a focal length f. A real object is placed at a distance f in front of it from the pole produces an image at

Options

A. Infinity
B. f
C. f/2
D. 2f

Answer

To solve the problem, we will use the mirror formula and the properties of convex mirrors. Here’s a step-by-step solution:

### Step 1: Understand the given information
We have a convex mirror with a focal length \( f \). A real object is placed at a distance \( f \) in front of the mirror.

### Step 2: Set up the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) = focal length of the mirror (positive for convex mirrors)
- \( v \) = image distance (to be determined)
- \( u \) = object distance (negative for real objects)

### Step 3: Assign values to the variables
Since the object is placed at a distance \( f \) in front of the mirror, we have:
- \( u = -f \) (negative because the object is in front of the mirror)
- \( f = +f \) (positive for a convex mirror)

### Step 4: Substitute the values into the mirror formula
Now substituting the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{-f} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{f} \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{f} \]
\[ \frac{1}{v} = \frac{2}{f} \]

### Step 6: Solve for \( v \)
Taking the reciprocal to find \( v \):
\[ v = \frac{f}{2} \]

### Step 7: Determine the nature of the image
Since \( v \) is positive, the image is formed on the same side as the object (which is characteristic of virtual images in convex mirrors). The image will be located between the focal point and the pole of the mirror.

### Conclusion
Thus, the image is formed at a distance \( \frac{f}{2} \) from the pole of the mirror, and it is virtual, erect, and diminished.

### Final Answer
The image is formed at a distance \( \frac{f}{2} \) from the pole of the convex mirror.

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