A luminous object is separated from a screen by distance d. A convex lends is placed between the object and the screeen such that it forms a distinct image on the screen. The maximum possible focal length of this convex lens is.

Options

A. `4d`
B. `2d`
C. `d//2`
D. `d//4`

Explanation

To solve the problem of finding the maximum possible focal length of a convex lens placed between a luminous object and a screen, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**: We have a luminous object and a screen separated by a distance \( d \). A convex lens is placed between them, forming a distinct image on the screen.

2. **Use the Lens Formula**: The lens formula relates the object distance \( u \), image distance \( v \), and focal length \( f \) of a lens:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

3. **Define Distances**: Let the distance from the object to the lens be \( u \) and from the lens to the screen be \( v \). Thus, we have:
\[ u + v = d \]

4. **Express \( v \) in terms of \( u \)**: From the equation \( u + v = d \), we can express \( v \) as:
\[ v = d - u \]

5. **Substitute \( v \) in the Lens Formula**: Now, substituting \( v \) into the lens formula gives:
\[ \frac{1}{f} = \frac{1}{d - u} - \frac{1}{u} \]

6. **Combine the Fractions**: To combine the fractions on the right side, we find a common denominator:
\[ \frac{1}{f} = \frac{u - (d - u)}{u(d - u)} = \frac{2u - d}{u(d - u)} \]

7. **Rearranging for \( f \)**: Inverting the equation gives:
\[ f = \frac{u(d - u)}{2u - d} \]

8. **Maximize \( f \)**: To find the maximum possible focal length, we need to find the critical points of \( f \) with respect to \( u \). We can differentiate \( f \) with respect to \( u \) and set the derivative to zero. However, a simpler approach is to analyze the function:
- The maximum value of \( f \) occurs when \( u = \frac{d}{2} \) (the lens is placed at the midpoint between the object and the screen).

9. **Substituting \( u = \frac{d}{2} \)**: Now substituting \( u = \frac{d}{2} \) into the equation for \( f \):
\[ f = \frac{\frac{d}{2} \left(d - \frac{d}{2}\right)}{2 \cdot \frac{d}{2} - d} = \frac{\frac{d}{2} \cdot \frac{d}{2}}{0} \]
However, we need to ensure \( 2u \neq d \) to avoid division by zero.

10. **Final Calculation**: The maximum focal length can be derived from the displacement method as:
\[ f = \frac{d^2}{4d} = \frac{d}{4} \]

### Conclusion:
The maximum possible focal length of the convex lens is:
\[ \boxed{\frac{d}{4}} \]