In an ellipse the distance between the foci is 8 and the distance between the directrices is 25, then the ratio of the length of major and minor axis is

Options

A. `(5)/(sqrt(17))`
B. `(3)/(sqrt(17))`
C. `(4)/(sqrt(17))`
D. `(6)/(sqrt(17))`

Answer

To solve the problem, we need to find the ratio of the lengths of the major and minor axes of an ellipse given the distance between the foci and the distance between the directrices.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The distance between the foci is 8.
- The distance between the directrices is 25.

2. **Use the Relationship for the Distance Between Foci:**
- The distance between the foci of an ellipse is given by \( 2c \), where \( c = ae \) (with \( e \) being the eccentricity).
- Therefore, we have:
\[ 2c = 8 \implies c = 4 \]

3. **Use the Relationship for the Distance Between Directrices:**
- The distance between the directrices is given by \( \frac{2a}{e} \).
- Therefore, we have:
\[ \frac{2a}{e} = 25 \implies 2a = 25e \implies a = \frac{25e}{2} \]

4. **Relate \( c \), \( a \), and \( b \):**
- We know that \( c^2 = a^2 - b^2 \).
- Substituting \( c = 4 \):
\[ 4^2 = a^2 - b^2 \implies 16 = a^2 - b^2 \implies b^2 = a^2 - 16 \]

5. **Substitute \( a \) in terms of \( e \):**
- From step 3, substitute \( a = \frac{25e}{2} \) into the equation for \( b^2 \):
\[ b^2 = \left(\frac{25e}{2}\right)^2 - 16 \]
\[ b^2 = \frac{625e^2}{4} - 16 \]

6. **Substitute \( c \) in terms of \( a \) and \( e \):**
- We know \( c = ae \), hence:
\[ 4 = \left(\frac{25e}{2}\right)e \implies 4 = \frac{25e^2}{2} \implies 8 = 25e^2 \implies e^2 = \frac{8}{25} \implies e = \frac{2\sqrt{2}}{5} \]

7. **Substitute \( e \) back to find \( a \):**
- Now substitute \( e \) back into the equation for \( a \):
\[ a = \frac{25 \cdot \frac{2\sqrt{2}}{5}}{2} = 5\sqrt{2} \]

8. **Calculate \( b^2 \):**
- Substitute \( a \) into the equation for \( b^2 \):
\[ b^2 = (5\sqrt{2})^2 - 16 = 50 - 16 = 34 \implies b = \sqrt{34} \]

9. **Find the Ratio of Major and Minor Axes:**
- The lengths of the major and minor axes are \( 2a \) and \( 2b \) respectively.
- Therefore, the ratio is:
\[ \frac{2a}{2b} = \frac{a}{b} = \frac{5\sqrt{2}}{\sqrt{34}} = \frac{5}{\sqrt{17}} \]

### Final Answer:
The ratio of the length of the major axis to the minor axis is \( \frac{5}{\sqrt{17}} \).