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 determine the value of \( \frac{d}{2} \) where \( d \) is the sum of the distances from a point \( P \) on the ellipse to its two foci.

### Step-by-Step Solution:

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

2. **Calculate \( a \) and \( b \)**:
From \( a^2 = 25 \), we find \( a = \sqrt{25} = 5 \).
From \( b^2 = 9 \), we find \( b = \sqrt{9} = 3 \).

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

4. **Use the Property of Ellipses**:
A fundamental property of ellipses states that for any point \( P \) on the ellipse, the sum of the distances from \( P \) to the two foci is constant and equal to \( 2a \).
\[ d = P S_1 + P S_2 = 2a. \]

5. **Calculate \( d \)**:
Substituting the value of \( a \):
\[ d = 2 \times 5 = 10. \]

6. **Find \( \frac{d}{2} \)**:
Now, we need to find \( \frac{d}{2} \):
\[ \frac{d}{2} = \frac{10}{2} = 5. \]

### Final Answer:
Thus, the value of \( \frac{d}{2} \) is \( 5 \).

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