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 step-by-step, we will use the properties of an ellipse and the relationships between its parameters.

### Step 1: Identify the given information
We know that:
- The distance between the foci (2ae) = 8
- The distance between the directrices (2a/e) = 25

### Step 2: Set up equations based on the given information
From the distance between the foci:
\[ 2ae = 8 \]
This simplifies to:
\[ ae = 4 \quad \text{(1)} \]

From the distance between the directrices:
\[ 2\frac{a}{e} = 25 \]
This simplifies to:
\[ \frac{a}{e} = 12.5 \quad \text{(2)} \]

### Step 3: Multiply equations (1) and (2)
Now, we will multiply equation (1) and equation (2):
\[ (ae) \left(\frac{a}{e}\right) = 4 \cdot 12.5 \]
This simplifies to:
\[ a^2 = 50 \]
Thus, we find:
\[ a = \sqrt{50} = 5\sqrt{2} \]

### Step 4: Find eccentricity (e)
From equation (1):
\[ ae = 4 \]
Substituting \( a = 5\sqrt{2} \):
\[ 5\sqrt{2} \cdot e = 4 \]
Thus, we find:
\[ e = \frac{4}{5\sqrt{2}} \]

### Step 5: Find b using the relationship between a, b, and e
We know that:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Squaring both sides:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Substituting \( e \):
\[ \left(\frac{4}{5\sqrt{2}}\right)^2 = 1 - \frac{b^2}{(5\sqrt{2})^2} \]
Calculating \( e^2 \):
\[ \frac{16}{50} = 1 - \frac{b^2}{50} \]
Thus:
\[ \frac{16}{50} = 1 - \frac{b^2}{50} \]
Rearranging gives:
\[ \frac{b^2}{50} = 1 - \frac{16}{50} = \frac{34}{50} \]
Thus:
\[ b^2 = 34 \]
So:
\[ b = \sqrt{34} \]

### Step 6: Find the lengths of the major and minor axes
The lengths of the axes are:
- Major axis = \( 2a = 2 \cdot 5\sqrt{2} = 10\sqrt{2} \)
- Minor axis = \( 2b = 2\sqrt{34} \)

### Step 7: Calculate the ratio of the lengths of the major and minor axes
The ratio of the major axis to the minor axis is:
\[ \frac{2a}{2b} = \frac{a}{b} = \frac{5\sqrt{2}}{\sqrt{34}} \]

### Final Step: Simplify the ratio
To simplify:
\[ \frac{5\sqrt{2}}{\sqrt{34}} = \frac{5\sqrt{2}}{\sqrt{2 \cdot 17}} = \frac{5}{\sqrt{17}} \]

### Conclusion
The ratio of the length of the major axis to the minor axis is:
\[ \frac{5}{\sqrt{17}} \]

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