Consider any point P on the ellipse `(x^(2))/(25)+(y^(2))/(9)=1` in the first quadrant. Let r and s represent its distances from (4, 0) and (-4, 0) respectively, then (r + s) is equal to:

Options

A. 10 unit
B. 9 unit
C. 8 unit
D. 6 unit

Answer

To solve the problem, we need to find the sum of the distances \( r \) and \( s \) from the point \( P \) on the ellipse to the points \( (4, 0) \) and \( (-4, 0) \).

### Step-by-Step Solution:

1. **Identify the equation of the ellipse**:
The equation given is
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
This is the standard form of an ellipse centered at the origin.

2. **Determine the semi-major and semi-minor axes**:
From the equation, we can identify:
- \( a^2 = 25 \) which gives \( a = 5 \)
- \( b^2 = 9 \) which gives \( b = 3 \)

3. **Recognize the properties of the ellipse**:
For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant. The foci of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \), where \( c = \sqrt{a^2 - b^2} \).

4. **Calculate the distance \( c \)**:
\[ c = \sqrt{25 - 9} = \sqrt{16} = 4 \]
Thus, the foci are at \( (4, 0) \) and \( (-4, 0) \).

5. **Use the property of the ellipse**:
The property states that for any point \( P \) on the ellipse, the sum of the distances \( r \) and \( s \) from \( P \) to the foci is given by:
\[ r + s = 2a \]
Since we have already determined \( a = 5 \):
\[ r + s = 2 \times 5 = 10 \]

6. **Conclusion**:
Therefore, the value of \( r + s \) is
\[ \boxed{10} \]