If the distance between the foci and the distance between the two directricies of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` are in the ratio 3:2, then `b : a` is (a)`1:sqrt(2)` (b) `sqrt(3):sqrt(2)` (c)`1:2` (d) `2:1`

Options

A. `1:sqrt2`
B. `sqrt3:sqrt2`
C. `1:2`
D. `2:1`

Explanation

To solve the problem, we need to find the ratio \( b : a \) given that the distance between the foci and the distance between the two directrices of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are in the ratio \( 3:2 \).

### Step-by-Step Solution:

1. **Identify the distances**:
- The distance between the foci of the hyperbola is given by \( 2ae \), where \( e \) is the eccentricity.
- The distance between the directrices is given by \( \frac{2a}{e} \).

2. **Set up the ratio**:
- According to the problem, the ratio of the distance between the foci to the distance between the directrices is \( 3:2 \).
- Therefore, we can write:
\[ \frac{2ae}{\frac{2a}{e}} = \frac{3}{2} \]

3. **Simplify the ratio**:
- Simplifying the left-hand side:
\[ \frac{2ae \cdot e}{2a} = e^2 \]
- Thus, we have:
\[ e^2 = \frac{3}{2} \]

4. **Relate eccentricity to \( a \) and \( b \)**:
- The eccentricity \( e \) for a hyperbola is defined as:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
- Substituting \( e^2 = \frac{3}{2} \) into this equation gives:
\[ \frac{3}{2} = 1 + \frac{b^2}{a^2} \]

5. **Solve for \( \frac{b^2}{a^2} \)**:
- Rearranging the equation:
\[ \frac{b^2}{a^2} = \frac{3}{2} - 1 = \frac{1}{2} \]

6. **Find the ratio \( b : a \)**:
- Taking the square root of both sides:
\[ \frac{b}{a} = \frac{1}{\sqrt{2}} \]
- Therefore, the ratio \( b : a \) is:
\[ b : a = 1 : \sqrt{2} \]

### Final Answer:
The ratio \( b : a \) is \( 1 : \sqrt{2} \), which corresponds to option (a).