The distance between the foci is `4sqrt(13)` and the length of conjugate axis is 8 then, the eccentricity of the hyperbola is

Options

A. `sqrt(13)//3`
B. `sqrt(13)//5`
C. `sqrt(13)//7`
D. none

Answer

To find the eccentricity of the hyperbola given the distance between the foci and the length of the conjugate axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Distance between the foci = \( 4\sqrt{13} \)
- Length of the conjugate axis = 8

2. **Relate the distance between the foci to \( ae \):**
- The distance between the foci of a hyperbola is given by \( 2ae \).
- Therefore, we have:
\[ 2ae = 4\sqrt{13} \]
- Dividing both sides by 2:
\[ ae = 2\sqrt{13} \]

3. **Square both sides to find \( a^2e^2 \):**
- Squaring \( ae = 2\sqrt{13} \):
\[ a^2e^2 = (2\sqrt{13})^2 = 4 \times 13 = 52 \]
- This gives us our first equation:
\[ a^2e^2 = 52 \quad \text{(Equation 1)} \]

4. **Find the value of \( b \):**
- The length of the conjugate axis is given by \( 2b \).
- Thus, we have:
\[ 2b = 8 \implies b = 4 \]

5. **Calculate \( b^2 \):**
- Now, we compute \( b^2 \):
\[ b^2 = 4^2 = 16 \]

6. **Use the relationship between \( a^2 \), \( b^2 \), and \( e^2 \):**
- For hyperbolas, we have the relationship:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
- Rearranging gives:
\[ a^2e^2 = a^2 + b^2 \]

7. **Substituting known values into the equation:**
- From Equation 1, we substitute \( a^2e^2 = 52 \) and \( b^2 = 16 \):
\[ 52 = a^2 + 16 \]

8. **Solve for \( a^2 \):**
- Rearranging gives:
\[ a^2 = 52 - 16 = 36 \]

9. **Find the eccentricity \( e \):**
- Now we can substitute \( a^2 \) and \( b^2 \) back into the eccentricity formula:
\[ e^2 = 1 + \frac{b^2}{a^2} = 1 + \frac{16}{36} \]
- Simplifying gives:
\[ e^2 = 1 + \frac{4}{9} = \frac{9}{9} + \frac{4}{9} = \frac{13}{9} \]
- Taking the square root:
\[ e = \sqrt{\frac{13}{9}} = \frac{\sqrt{13}}{3} \]

### Final Answer:
The eccentricity of the hyperbola is \( \frac{\sqrt{13}}{3} \).

---