Assertion: Minimum distance between object and its real image by a convex lens is `4f.`
Reason: If object distance from a convex lens is `2f,` then its image distace is also `2f.`

Options

A. a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. b) If both Assertion and Reason are true but Reason is not the correct explanation of the Assertion.
C. c) If both Assertion is true, but the Reason is false.
D. d) If both Assertion is false but the Reason is true.

Answer

To solve the question, we need to analyze the assertion and reason provided regarding the behavior of a convex lens.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that the minimum distance between an object and its real image formed by a convex lens is `4f`, where `f` is the focal length of the lens.

2. **Using the Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Here, `u` is the object distance (negative in sign convention), and `v` is the image distance (positive for real images).

3. **Setting Up the Problem**:
- Let the object distance be `u = -x` (since object distance is negative) and the image distance be `v = d - x` (where `d` is the total distance between the object and image).
- Rearranging the lens formula gives:
\[ \frac{1}{f} = \frac{1}{d - x} + \frac{1}{x} \]

4. **Finding the Relationship**:
- By finding a common denominator and simplifying, we get:
\[ \frac{1}{f} = \frac{x + (d - x)}{x(d - x)} = \frac{d}{x(d - x)} \]
- This leads to:
\[ x(d - x) = fd \]
- Rearranging gives us a quadratic equation:
\[ x^2 - dx + fd = 0 \]

5. **Applying the Discriminant Condition**:
- For real roots, the discriminant of the quadratic must be non-negative:
\[ b^2 - 4ac \geq 0 \]
- Here, \( b = -d, a = 1, c = fd \):
\[ d^2 - 4fd \geq 0 \]
- Factoring gives:
\[ d(d - 4f) \geq 0 \]
- This implies that:
\[ d \geq 4f \]
- Thus, the minimum distance \( d \) between the object and the image is indeed \( 4f \).

6. **Understanding the Reason**:
- The reason states that if the object distance is \( 2f \), then the image distance is also \( 2f \).
- Substituting \( u = -2f \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{2f} \]
- This leads to:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{2f} = \frac{2 - 1}{2f} = \frac{1}{2f} \]
- Thus, \( v = 2f \), confirming that the image distance is indeed \( 2f \).

7. **Conclusion**:
- Both the assertion and reason are correct, and the reason correctly explains the assertion. Therefore, the answer to the question is that both the assertion and reason are true.

### Final Answer:
Both the assertion and reason are correct, and the reason is the correct explanation for the assertion.

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