Find the eccentricity of an ellipse beween whose foci is 10 and that between focus and corresponding directirx is 15

Explanation

To find the eccentricity of the ellipse given the distance between its foci and the distance between the focus and the corresponding directrix, we can follow these steps:

### Step 1: Understand the given information
We know:
- The distance between the foci (2c) = 10
- The distance between the focus and the corresponding directrix (a/e) = 15

### Step 2: Relate the distance between foci to 'c'
The distance between the foci of an ellipse is given by:
\[ 2c = 10 \]
From this, we can find 'c':
\[ c = \frac{10}{2} = 5 \]

### Step 3: Relate 'c' to 'a' and 'e'
We know that:
\[ c = ae \]
So we can express 'a' in terms of 'e':
\[ a = \frac{c}{e} = \frac{5}{e} \]

### Step 4: Use the distance to the directrix
The distance from the focus to the directrix is given by:
\[ \frac{a}{e} = 15 \]
Substituting \( a \) from the previous step:
\[ \frac{5/e}{e} = 15 \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{5}{e^2} = 15 \]

### Step 6: Solve for 'e'
Cross-multiplying gives:
\[ 5 = 15e^2 \]
Dividing both sides by 15:
\[ e^2 = \frac{5}{15} = \frac{1}{3} \]
Taking the square root:
\[ e = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

### Final Result
Thus, the eccentricity \( e \) of the ellipse is:
\[ e = \frac{1}{\sqrt{3}} \]