There are two vectors having magnitudes a and b (a `gt` b) . Ratio of their maximum possible resultant to that with minimum possible resultant is 3 . Choose the correct option .

Options

A. `a=2b`
B. `a=3b`
C. `a=4b`
D. `b=2a`

Explanation

To solve the problem, we need to find the relationship between the magnitudes of two vectors \( a \) and \( b \) given that the ratio of their maximum possible resultant to the minimum possible resultant is 3.

### Step-by-Step Solution:

1. **Understanding the Resultant of Two Vectors**:
The resultant \( R \) of two vectors \( a \) and \( b \) can be expressed using the formula:
\[ R = \sqrt{a^2 + b^2 + 2ab \cos \theta} \]
where \( \theta \) is the angle between the two vectors.

2. **Finding Maximum Resultant**:
The maximum resultant \( R_{\text{max}} \) occurs when \( \cos \theta = 1 \) (i.e., when the vectors are in the same direction):
\[ R_{\text{max}} = a + b \]

3. **Finding Minimum Resultant**:
The minimum resultant \( R_{\text{min}} \) occurs when \( \cos \theta = -1 \) (i.e., when the vectors are in opposite directions):
\[ R_{\text{min}} = |a - b| = a - b \quad \text{(since \( a > b \))} \]

4. **Setting Up the Ratio**:
According to the problem, the ratio of the maximum resultant to the minimum resultant is given as:
\[ \frac{R_{\text{max}}}{R_{\text{min}}} = 3 \]
Substituting the expressions for \( R_{\text{max}} \) and \( R_{\text{min}} \):
\[ \frac{a + b}{a - b} = 3 \]

5. **Cross-Multiplying**:
Cross-multiplying gives:
\[ a + b = 3(a - b) \]

6. **Expanding and Rearranging**:
Expanding the right side:
\[ a + b = 3a - 3b \]
Rearranging the equation:
\[ a + b + 3b = 3a \]
\[ 4b = 2a \]

7. **Final Relationship**:
Dividing both sides by 2:
\[ a = 2b \]

### Conclusion:
The relationship between the magnitudes of the vectors is \( a = 2b \).