The distance between the directrix is equal to 8 times the distance between the foci then e =

Options

A. `1/2`
B. `(1)/(2sqrt(2))`
C. `1/4`
D. `1/8`

Answer

To solve the problem, we need to find the eccentricity \( e \) of the ellipse given that the distance between the directrices is equal to 8 times the distance between the foci.

### Step-by-step Solution:

1. **Understand the Relationships**:
- The distance between the directrices of an ellipse is given by the formula:
\[ \text{Distance between directrices} = \frac{2A}{e} \]
- The distance between the foci of an ellipse is given by:
\[ \text{Distance between foci} = 2Ae \]

2. **Set Up the Equation**:
- According to the problem, the distance between the directrices is equal to 8 times the distance between the foci. Therefore, we can set up the equation:
\[ \frac{2A}{e} = 8 \times (2Ae) \]

3. **Simplify the Equation**:
- Simplifying the right side:
\[ \frac{2A}{e} = 16Ae \]
- Now, we can eliminate \( 2A \) from both sides (assuming \( A \neq 0 \)):
\[ \frac{1}{e} = 8e \]

4. **Multiply Both Sides by \( e \)**:
- To eliminate the fraction, multiply both sides by \( e \):
\[ 1 = 8e^2 \]

5. **Solve for \( e^2 \)**:
- Rearranging gives:
\[ e^2 = \frac{1}{8} \]

6. **Find \( e \)**:
- Taking the square root of both sides:
\[ e = \sqrt{\frac{1}{8}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} \]

### Final Answer:
Thus, the eccentricity \( e \) is:
\[ e = \frac{1}{2\sqrt{2}} \]