If P is a point on the ellipse `(x^(2))/(36)+(y^(2))/(9)=1`, S and S ’ are the foci of the ellipse then find `SP + S^1P`

Answer

To solve the problem, we need to find the sum of the distances from a point \( P \) on the ellipse to its foci \( S \) and \( S' \).

### Step 1: Identify the equation of the ellipse
The equation given is:
\[ \frac{x^2}{36} + \frac{y^2}{9} = 1 \]
From this, we can identify \( a^2 = 36 \) and \( b^2 = 9 \).

### Step 2: Calculate the values of \( a \) and \( b \)
Taking the square roots:
\[ a = \sqrt{36} = 6 \quad \text{and} \quad b = \sqrt{9} = 3 \]

### Step 3: Determine the foci of the ellipse
The foci \( S \) and \( S' \) of the ellipse are located at \( (c, 0) \) and \( (-c, 0) \) respectively, where \( c \) is given by:
\[ c = \sqrt{a^2 - b^2} = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3} \]
Thus, the coordinates of the foci are:
\[ S(3\sqrt{3}, 0) \quad \text{and} \quad S'(-3\sqrt{3}, 0) \]

### Step 4: Use the property of the ellipse
A key property of ellipses states that for any point \( P \) on the ellipse, the sum of the distances from \( P \) to the two foci \( S \) and \( S' \) is constant and equal to the length of the major axis.

### Step 5: Calculate the length of the major axis
The length of the major axis is given by:
\[ 2a = 2 \times 6 = 12 \]

### Conclusion
Therefore, the sum of the distances \( SP + S'P \) is:
\[ SP + S'P = 12 \]

### Final Answer
\[ SP + S'P = 12 \]