If the distance between foci of a hyperbola is twice the distance between its directrices, then the eccentricity of conjugate hyperbola is :

Options

A. 2
B. ` sqrt 3 `
C. `sqrt2 `
D. ` ( 3 ) /(2) `

Explanation

To solve the problem, we need to find the eccentricity of the conjugate hyperbola given that the distance between the foci of a hyperbola is twice the distance between its directrices.

### Step-by-Step Solution:

1. **Understanding the Distances**:
- The distance between the foci of a hyperbola is given by \(2ae\), where \(a\) is the semi-major axis and \(e\) is the eccentricity of the hyperbola.
- The distance between the directrices of a hyperbola is given by \(\frac{2a}{e}\).

2. **Setting Up the Equation**:
- According to the problem, the distance between the foci is twice the distance between the directrices:
\[ 2ae = 2 \left(\frac{2a}{e}\right) \]

3. **Simplifying the Equation**:
- We can simplify the equation:
\[ 2ae = \frac{4a}{e} \]
- Dividing both sides by \(2a\) (assuming \(a \neq 0\)):
\[ e = \frac{2}{e} \]

4. **Multiplying Both Sides by \(e\)**:
- To eliminate the fraction, multiply both sides by \(e\):
\[ e^2 = 2 \]

5. **Finding the Eccentricity**:
- Taking the square root of both sides gives:
\[ e = \sqrt{2} \]

6. **Conjugate Hyperbola Eccentricity**:
- For the conjugate hyperbola, the eccentricity \(e'\) is given by:
\[ e' = \sqrt{\frac{a^2 + b^2}{b^2}} \]
- From the relationship \(e^2 = \frac{a^2 + b^2}{a^2}\), we have:
\[ e^2 = 2 \implies a^2 + b^2 = 2a^2 \implies b^2 = a^2 \]
- Therefore, substituting \(b^2 = a^2\) into the eccentricity formula for the conjugate hyperbola:
\[ e' = \sqrt{\frac{a^2 + a^2}{a^2}} = \sqrt{2} \]

### Final Answer:
Thus, the eccentricity of the conjugate hyperbola is \(\sqrt{2}\).