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 given the distance between the object and the screen \( x \) and the magnification \( m \).

### Step-by-Step Solution:

1. **Understanding the Variables**:
- 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 by \( x \), so we can write:
\[ u + v = x \quad \text{(1)} \]

2. **Using Magnification**:
- The magnification \( m \) produced by the lens is defined as:
\[ m = \frac{h'}{h} = \frac{v}{u} \quad \text{(where \( h' \) is the height of the image and \( h \) is the height of the object)} \]
- Rearranging this gives:
\[ v = m \cdot u \quad \text{(2)} \]

3. **Substituting for \( v \)**:
- Substitute equation (2) into equation (1):
\[ u + m \cdot u = x \]
- Factor out \( u \):
\[ u(1 + m) = x \]
- Solving for \( u \) gives:
\[ u = \frac{x}{1 + m} \quad \text{(3)} \]

4. **Finding \( v \)**:
- Now substitute equation (3) back into equation (2) to find \( v \):
\[ v = m \cdot u = m \cdot \frac{x}{1 + m} = \frac{mx}{1 + m} \quad \text{(4)} \]

5. **Using the Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Substitute equations (3) and (4) into the lens formula:
\[ \frac{1}{f} = \frac{1 + m}{mx} - \frac{1 + m}{x} \]

6. **Finding a Common Denominator**:
- The common denominator for the right-hand side is \( mx \):
\[ \frac{1}{f} = \frac{(1 + m) - (1 + m)m}{mx} \]
- Simplifying gives:
\[ \frac{1}{f} = \frac{1 + m - m - m^2}{mx} = \frac{1 - m^2}{mx} \]

7. **Reciprocal to Find \( f \)**:
- Taking the reciprocal to find \( f \):
\[ f = \frac{mx}{1 - m^2} \]

8. **Final Expression for Focal Length**:
- Thus, the focal length \( f \) of the lens is:
\[ f = \frac{mx}{(1 + m)^2} \]