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

Options

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

Explanation

To find the distance of the object from a concave mirror that produces a real image `n` times the size of the object, we can follow these steps:

### Step 1: Understand the relationship between object distance (U), image distance (V), and magnification (M).
The magnification (M) for mirrors is given by the formula:
\[ M = -\frac{V}{U} \]
Since the image is real and `n` times the size of the object, we can express this as:
\[ M = -n \]
This leads to:
\[ -n = -\frac{V}{U} \]
Thus, we can rearrange this to find:
\[ V = nU \]

### Step 2: Use the mirror formula.
The mirror formula relates the object distance (U), image distance (V), and the focal length (f) of the mirror:
\[ \frac{1}{f} = \frac{1}{V} + \frac{1}{U} \]

### Step 3: Substitute the expression for V into the mirror formula.
Substituting \( V = nU \) into the mirror formula gives:
\[ \frac{1}{f} = \frac{1}{nU} + \frac{1}{U} \]

### Step 4: Simplify the equation.
Finding a common denominator for the right side:
\[ \frac{1}{f} = \frac{1 + n}{nU} \]
Now, we can rearrange this to solve for U:
\[ nU = (1 + n)f \]
Thus:
\[ U = \frac{(1 + n)f}{n} \]

### Step 5: Determine the sign of U.
Since the object distance (U) for a concave mirror is taken as negative in the sign convention, we have:
\[ U = -\frac{(1 + n)f}{n} \]

### Final Answer:
The distance of the object from the mirror is:
\[ U = -\frac{(1 + n)f}{n} \]

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