Statement-1: The circle of smallest radius passing through two given point A and B must be of radius 1/2 AB.
Statement -2: A straight line is a shortest distance between two points .

Options

A. Statement-1 is True , Statement -2 is true , Statement -2 is a correct explanation for Statement-1
B. Statement-1 is True , Statement -2 is true , Statement -2 is NOT a correct explanation for Statement-1
C. Statement -1 is True , Statement -2 is False
D. Statement-1 is False , Statement-2 is True

Answer

To analyze the statements given in the question, we will break down each statement step by step and provide a logical explanation for each.

### Step-by-Step Solution:

**Step 1: Understanding Statement 1**
- Statement 1 claims that the circle of the smallest radius passing through two given points A and B must have a radius of \( \frac{1}{2} AB \).
- To analyze this, we need to recognize that the smallest circle that can pass through two points A and B will have its diameter equal to the distance between A and B (denoted as AB).

**Hint for Step 1:** Consider the geometric properties of circles and how they relate to chords and diameters.

**Step 2: Establishing the Diameter**
- The diameter of the circle that passes through points A and B is simply the distance between these two points, which is \( AB \).
- Therefore, the radius of this circle is given by the formula for the radius in terms of the diameter:
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{AB}{2} \]

**Hint for Step 2:** Recall the relationship between the diameter and radius of a circle.

**Step 3: Conclusion for Statement 1**
- Since we have established that the radius of the circle passing through points A and B is indeed \( \frac{AB}{2} \), Statement 1 is correct.

**Hint for Step 3:** Verify the calculations and the relationship between radius and diameter.

**Step 4: Understanding Statement 2**
- Statement 2 states that a straight line is the shortest distance between two points.
- This is a well-known geometric principle. If you take any two points A and B, the straight line connecting them is the shortest path compared to any other possible path (curved or otherwise).

**Hint for Step 4:** Think about the definition of distance in geometry and how it applies to straight lines versus curves.

**Step 5: Conclusion for Statement 2**
- Since the straight line indeed represents the shortest distance between two points, Statement 2 is also correct.

**Hint for Step 5:** Consider examples of paths between two points to reinforce the concept.

### Final Conclusion:
- Both statements are correct, but Statement 2 does not provide a correct explanation for Statement 1. Thus, the final answer is that Statement 1 is true, and Statement 2 is also true, but they are independent of each other.