A concave mirror of focal length `f` produces a real image `n` times the size of the object. What is the distance of the object from the mirror?

Options

A. (n - 1)f
B. `{((n-1))/n}f`
C. `{((n+1))/n}f`
D. (n + 1)f

Answer

To solve the problem step by step, we will use the concepts of concave mirrors, magnification, and the mirror formula.

### Step 1: Understand the given information
We have a concave mirror with a focal length \( f \) (which is negative). The mirror produces a real image that is \( n \) times the size of the object. Since the image is real and inverted, the magnification \( m \) is given by:
\[ m = -n \]

### Step 2: Relate magnification to object and image distances
Magnification \( m \) is also defined as the ratio of the image distance \( v \) to the object distance \( u \):
\[ m = -\frac{v}{u} \]
Substituting the expression for magnification, we have:
\[ -n = -\frac{v}{u} \]
This simplifies to:
\[ v = n u \]

### Step 3: Use the mirror formula
The mirror formula for a concave mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( v = n u \) into the mirror formula, we get:
\[ \frac{1}{f} = \frac{1}{n u} + \frac{1}{u} \]

### Step 4: Combine the fractions
To combine the fractions on the right side, we find a common denominator:
\[ \frac{1}{f} = \frac{1 + n}{n u} \]
This can be rewritten as:
\[ \frac{1}{f} = \frac{n + 1}{n u} \]

### Step 5: Solve for \( u \)
Now, we can rearrange the equation to solve for \( u \):
\[ u = \frac{f(n + 1)}{n} \]

### Conclusion
Thus, the distance of the object from the mirror is:
\[ u = \frac{f(n + 1)}{n} \]