If P is any point on the ellipse `9x^(2) + 36y^(2) = 324` whose foci are S and S'. Then, SP + S' P equals

Options

A. 3
B. 12
C. 36
D. 324

Answer

To solve the problem, we need to find the sum of the distances from any point \( P \) on the ellipse \( 9x^2 + 36y^2 = 324 \) to its foci \( S \) and \( S' \).

### Step 1: Convert the equation of the ellipse to standard form
The given equation is:
\[ 9x^2 + 36y^2 = 324 \]
We divide the entire equation by 324:
\[ \frac{9x^2}{324} + \frac{36y^2}{324} = 1 \]
This simplifies to:
\[ \frac{x^2}{36} + \frac{y^2}{9} = 1 \]
Now, we can identify \( a^2 \) and \( b^2 \):
- \( a^2 = 36 \) so \( a = 6 \)
- \( b^2 = 9 \) so \( b = 3 \)

### Step 2: Identify the foci of the ellipse
For an ellipse, the foci \( S \) and \( S' \) can be found using the formula \( c = \sqrt{a^2 - b^2} \):
\[ c = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3} \]
The foci are located at \( (c, 0) \) and \( (-c, 0) \), which gives us:
- \( S = (3\sqrt{3}, 0) \)
- \( S' = (-3\sqrt{3}, 0) \)

### Step 3: Use the property of the ellipse
A fundamental property of ellipses states that for any point \( P \) on the ellipse, the sum of the distances from \( P \) to the foci \( S \) and \( S' \) is constant and equal to \( 2a \):
\[ SP + S'P = 2a \]
Substituting the value of \( a \):
\[ SP + S'P = 2 \times 6 = 12 \]

### Conclusion
Thus, the sum of the distances from point \( P \) to the foci \( S \) and \( S' \) is:
\[ SP + S'P = 12 \]