If P is a point on the ellipse `(X^(2))/(9) + (y^(2))/(4) =1 `
whose foci are S and S' then the value of PS + PS' is

Options

A. 6
B. 4
C. 3
D. 2

Answer

To solve the problem, we need to find the value of \( PS + PS' \) for a point \( P \) on the ellipse given by the equation:

\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

### Step 1: Identify the parameters of the ellipse
The standard form of the ellipse is given by:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

From the equation, we can identify:
- \( a^2 = 9 \) which gives \( a = 3 \)
- \( b^2 = 4 \) which gives \( b = 2 \)

### Step 2: Determine the foci of the ellipse
The foci \( S \) and \( S' \) of the ellipse can be found using the formula:

\[ c = \sqrt{a^2 - b^2} \]

Calculating \( c \):

\[ c = \sqrt{9 - 4} = \sqrt{5} \]

Thus, the coordinates of the foci are:
- \( S(-\sqrt{5}, 0) \)
- \( S'(\sqrt{5}, 0) \)

### 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 \( 2a \).

### Step 4: Calculate \( PS + PS' \)
Since we have already established that \( a = 3 \):

\[ PS + PS' = 2a = 2 \times 3 = 6 \]

### Final Answer
Thus, the value of \( PS + PS' \) is:

\[ \boxed{6} \]

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