A concave mirror of focal length produces an image m times the size of the object. If image is real, then the distance of the object from mirror is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have a concave mirror with a focal length \( f \).
- The magnification \( m \) is given, and it is stated that the image is real.

2. **Write the Magnification Formula**:
- The magnification \( m \) for mirrors is given by the formula:
\[ m = -\frac{v}{u} \]
- Since the image is real, the magnification will be negative, so we can express it as:
\[ m = -\frac{v}{u} \implies v = -mu \]

3. **Use the Mirror Formula**:
- The mirror formula for a concave mirror is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting \( v = -mu \) into the mirror formula gives:
\[ \frac{1}{f} = \frac{1}{-mu} + \frac{1}{u} \]

4. **Simplify the Equation**:
- Rewrite the equation:
\[ \frac{1}{f} = -\frac{1}{mu} + \frac{1}{u} \]
- Find a common denominator for the right side:
\[ \frac{1}{f} = \frac{-1 + m}{mu} \]

5. **Cross Multiply to Solve for \( u \)**:
- Cross multiplying gives:
\[ u = \frac{f(-1 + m)}{1} \]
- This simplifies to:
\[ u = f(m - 1) \]

6. **Find the Magnitude of \( u \)**:
- Since we are interested in the magnitude of the object distance \( u \):
\[ |u| = f(m + 1) \]

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