A convex lens of focal length `f` is placed somewhere in between an object and a screen. The distance between the object and the screen is `x`. If the numerical value of the magnification produced by the lens is `m`, then the focal lnegth oof the lens is .

Options

A. `(mx)/((m+1)^(2))`
B. `(mx)/((m-1)^(2))`
C. `((m+1)^(2))/(m)x`
D. `((m-1)^(2))/(m)x`

Explanation

To solve the problem, we need to find the focal length \( f \) of a convex lens placed between an object and a screen, given the distance \( x \) between the object and the screen, and the magnification \( m \) produced by the lens.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let \( u \) be the object distance from the lens.
- Let \( v \) be the image distance from the lens.
- The total distance between the object and the screen is given as \( x \), so we have:
\[ u + v = x \]

2. **Using the Magnification Formula**:
- The magnification \( m \) produced by the lens is given by the formula:
\[ m = -\frac{v}{u} \]
- Rearranging this gives:
\[ v = -mu \]

3. **Substituting for \( v \)**:
- Substitute \( v \) in the equation \( u + v = x \):
\[ u - mu = x \]
- Factor out \( u \):
\[ u(1 - m) = x \]
- Thus, we can express \( u \) as:
\[ u = \frac{x}{1 - m} \]

4. **Finding \( v \)**:
- Now, substitute \( u \) back into the equation for \( v \):
\[ v = -m \left(\frac{x}{1 - m}\right) = \frac{-mx}{1 - m} \]

5. **Using the Lens Formula**:
- The lens formula relates the focal length \( f \), object distance \( u \), and image distance \( v \):
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Substitute the values of \( u \) and \( v \):
\[ \frac{1}{f} = \frac{1 - m}{-mx} - \frac{1 - m}{x} \]

6. **Finding a Common Denominator**:
- The common denominator for the right-hand side is \( -mx(1 - m) \):
\[ \frac{1}{f} = \frac{-(1 - m) + m(1 - m)}{-mx(1 - m)} \]
- Simplifying this gives:
\[ \frac{1}{f} = \frac{m(1 - m) - (1 - m)}{-mx(1 - m)} \]
- This simplifies to:
\[ \frac{1}{f} = \frac{(m - 1)(1 - m)}{-mx(1 - m)} \]

7. **Final Expression for \( f \)**:
- Rearranging gives:
\[ f = \frac{mx}{(1 + m)^2} \]

### Final Answer:
The focal length \( f \) of the lens is given by:
\[ f = \frac{mx}{(1 + m)^2} \]