A convex lens produces a real image `m` times the size of the object. What will be the distance of the object from the lens ?

Options

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

Answer

To solve the problem step by step, we will use the lens formula and the magnification formula for a convex lens.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have a convex lens that produces a real image which is `m` times the size of the object.
- Since the image is real, the magnification (m) will be negative. Therefore, we can write:
\[ m = -\frac{h'}{h} \quad \text{(where } h' \text{ is the height of the image and } h \text{ is the height of the object)} \]
- Thus, we have:
\[ -m = \frac{h'}{h} \]

2. **Use the Magnification Formula:**
- The magnification (m) for a lens can also be expressed in terms of the object distance (u) and the image distance (v):
\[ m = \frac{v}{u} \]
- Since the image is real, we can write:
\[ -m = \frac{v}{u} \quad \Rightarrow \quad v = -mu \]

3. **Apply the Lens Formula:**
- The lens formula relates the object distance (u), image distance (v), and focal length (f) of the lens:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
- Substituting \( v = -mu \) into the lens formula gives:
\[ \frac{1}{f} = \frac{1}{u} - \frac{1}{mu} \]

4. **Combine the Terms:**
- To combine the fractions on the right side:
\[ \frac{1}{f} = \frac{m - 1}{mu} \]

5. **Rearranging for u:**
- Rearranging the equation to solve for u:
\[ u = \frac{(m - 1)f}{m} \]

6. **Final Expression:**
- Therefore, the distance of the object from the lens is:
\[ u = \frac{(m - 1)f}{m} \]

### Conclusion:
The distance of the object from the lens is given by:
\[ u = \frac{(m - 1)f}{m} \]