A point P on the ellipse `(x^2)/(25)+(y^2)/(9)=1` has the eccentric angle `(pi)/(8)`. The sum of the distances of P from the two foci is d. Then `(d)/(2)` is equal to:

Answer

To solve the problem, we need to find the value of \( \frac{d}{2} \), where \( d \) is the sum of the distances from a point \( P \) on the ellipse to the two foci of the ellipse.

### Step-by-Step Solution:

1. **Identify the equation of the ellipse**:
The given ellipse is
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
Here, \( a^2 = 25 \) and \( b^2 = 9 \).

2. **Calculate the semi-major and semi-minor axes**:
- The semi-major axis \( a \) is given by \( a = \sqrt{25} = 5 \).
- The semi-minor axis \( b \) is given by \( b = \sqrt{9} = 3 \).

3. **Find the foci of the ellipse**:
The foci of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).
- Calculate \( c \):
\[ c = \sqrt{25 - 9} = \sqrt{16} = 4 \]
Therefore, the foci are at \( (4, 0) \) and \( (-4, 0) \).

4. **Use the eccentric angle to find the coordinates of point \( P \)**:
The eccentric angle \( \theta \) is given as \( \frac{\pi}{8} \). The coordinates of point \( P \) on the ellipse can be expressed as:
\[ P(x, y) = (a \cos \theta, b \sin \theta) \]
Substituting the values of \( a \), \( b \), and \( \theta \):
\[ P\left(5 \cos\left(\frac{\pi}{8}\right), 3 \sin\left(\frac{\pi}{8}\right)\right) \]

5. **Calculate the distances from point \( P \) to the foci**:
Let \( S_1 = (4, 0) \) and \( S_2 = (-4, 0) \).
- The distance \( S_1P \) is:
\[ S_1P = \sqrt{(5 \cos\left(\frac{\pi}{8}\right) - 4)^2 + (3 \sin\left(\frac{\pi}{8}\right))^2} \]
- The distance \( S_2P \) is:
\[ S_2P = \sqrt{(5 \cos\left(\frac{\pi}{8}\right) + 4)^2 + (3 \sin\left(\frac{\pi}{8}\right))^2} \]

6. **Use the property of ellipses**:
The sum of the distances from any point \( P \) on the ellipse to the two foci is a constant, equal to \( 2a \):
\[ d = S_1P + S_2P = 2a = 2 \times 5 = 10 \]

7. **Calculate \( \frac{d}{2} \)**:
\[ \frac{d}{2} = \frac{10}{2} = 5 \]

### Final Answer:
\[ \frac{d}{2} = 5 \]