Let P be a variable point on the ellipse with foci `S_(1)` and `S_(2)` . If A be the area of `trianglePS_(1)S_(2)` then find the maximum value of A

Answer

To find the maximum area \( A \) of triangle \( PS_1S_2 \) where \( P \) is a variable point on the ellipse with foci \( S_1 \) and \( S_2 \), we can follow these steps:

### Step 1: Understand the Ellipse
The standard equation of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

### Step 2: Parametrize the Point \( P \)
Let \( P \) be represented in parametric form as:
\[ P(a \cos \theta, b \sin \theta) \]
where \( \theta \) is a parameter that varies.

### Step 3: Identify the Foci
The foci \( S_1 \) and \( S_2 \) of the ellipse are located at:
\[ S_1(-c, 0) \quad \text{and} \quad S_2(c, 0) \]
where \( c = \sqrt{a^2 - b^2} \).

### Step 4: Area of Triangle \( PS_1S_2 \)
The area \( A \) of triangle \( PS_1S_2 \) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \( S_1S_2 \) is \( 2c \) and the height is the perpendicular distance from point \( P \) to the line segment \( S_1S_2 \).

### Step 5: Calculate the Height
The height from point \( P(a \cos \theta, b \sin \theta) \) to the line joining \( S_1 \) and \( S_2 \) (the x-axis) is simply the y-coordinate of point \( P \), which is \( b \sin \theta \).

### Step 6: Area Expression
Thus, the area \( A \) can be expressed as:
\[ A = \frac{1}{2} \times 2c \times (b \sin \theta) = c b \sin \theta \]
Substituting \( c = \sqrt{a^2 - b^2} \):
\[ A = b \sqrt{a^2 - b^2} \sin \theta \]

### Step 7: Maximizing the Area
To find the maximum area, we need to maximize \( A \). The maximum value of \( \sin \theta \) is 1, which occurs when \( \theta = \frac{\pi}{2} \). Thus, the maximum area \( A_{\text{max}} \) is:
\[ A_{\text{max}} = b \sqrt{a^2 - b^2} \]

### Conclusion
The maximum area of triangle \( PS_1S_2 \) is:
\[ A_{\text{max}} = b \sqrt{a^2 - b^2} \]