P is a variable point on the ellipse with foci `S_1` and `S_2`. If A is the area of the the triangle `PS_1S_2`, the maximum value of A is

Options

A. ab
B. abe
C. `(1)/(2) ab`
D. `(1)/(2) abe`

Explanation

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 and its foci
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. The foci \( S_1 \) and \( S_2 \) of the ellipse are located at \( (ae, 0) \) and \( (-ae, 0) \) respectively, where \( e = \sqrt{1 - \frac{b^2}{a^2}} \).

### Step 2: Parametrize the point \( P \)
Let the coordinates of point \( P \) on the ellipse be:
\[ P = (a \cos \theta, b \sin \theta) \]
where \( \theta \) is a parameter that varies.

### Step 3: Calculate the area of triangle \( PS_1S_2 \)
The area \( A \) of triangle \( PS_1S_2 \) can be calculated using the formula for the area of a triangle given by vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
- \( P = (a \cos \theta, b \sin \theta) \)
- \( S_1 = (ae, 0) \)
- \( S_2 = (-ae, 0) \)

The area becomes:
\[ A = \frac{1}{2} \left| a \cos \theta (0 - 0) + ae(0 - b \sin \theta) + (-ae)(b \sin \theta - 0) \right| \]
Simplifying this:
\[ A = \frac{1}{2} \left| -aeb \sin \theta + aeb \sin \theta \right| = \frac{1}{2} \left| 2aeb \sin \theta \right| = aeb \sin \theta \]

### Step 4: Maximize the area
To find the maximum area, we need to maximize \( A = aeb \sin \theta \). The maximum value of \( \sin \theta \) is 1, which occurs when \( \theta = \frac{\pi}{2} \).

### Step 5: Calculate the maximum area
Substituting \( \sin \theta = 1 \) into the area formula:
\[ A_{\text{max}} = aeb \cdot 1 = aeb \]

### Conclusion
The maximum area \( A \) of triangle \( PS_1S_2 \) is:
\[ \boxed{aeb} \]