To solve the problem, we need to find the maximum area of triangle PXY, where P is the point where the circle touches the line, X is a point on the circle, and Y is the foot of the perpendicular from X to the line.
### Step-by-Step Solution:
1. **Understanding the Circle and Line**:
- The circle C has a radius of 1 and touches the line L at point P. Since the circle touches the line at P, we can place the circle centered at the origin (0, 0) and the line L can be represented as \(y = 1\).
- The equation of the circle is \(x^2 + y^2 = 1\).
2. **Identifying Points**:
- The point P, where the circle touches the line, is at (0, 1).
- A point X on the circle can be represented in parametric form as \(X = (\cos \theta, \sin \theta)\), where \(\theta\) is the angle from the positive x-axis.
3. **Finding the Foot of the Perpendicular (Point Y)**:
- The foot of the perpendicular from point X to line L (y = 1) will have the same x-coordinate as X and a y-coordinate of 1. Therefore, the coordinates of Y will be \(Y = (\cos \theta, 1)\).
4. **Area of Triangle PXY**:
- The area \(A\) of triangle PXY can be calculated using the formula for the area of a triangle given by vertices:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
- Here, the base PY is the vertical distance from P to Y, which is \(1 - \sin \theta\), and the height PX is the horizontal distance from P to X, which is \(|\cos \theta|\).
- Therefore, the area can be expressed as:
\[
A = \frac{1}{2} \times |\cos \theta| \times (1 - \sin \theta)
\]
5. **Maximizing the Area**:
- To find the maximum area, we need to maximize the function:
\[
A(\theta) = \frac{1}{2} \cos \theta (1 - \sin \theta)
\]
- This can be simplified to:
\[
A(\theta) = \frac{1}{2} \left(\cos \theta - \cos \theta \sin \theta\right)
\]
6. **Finding the Derivative**:
- Differentiate \(A(\theta)\) with respect to \(\theta\):
\[
A'(\theta) = \frac{1}{2} \left(-\sin \theta + \sin^2 \theta - \cos^2 \theta\right)
\]
- Set \(A'(\theta) = 0\) to find critical points:
\[
-\sin \theta + \sin^2 \theta - \cos^2 \theta = 0
\]
7. **Solving the Equation**:
- Rearranging gives:
\[
2\sin^2 \theta - \sin \theta - 1 = 0
\]
- This can be factored or solved using the quadratic formula to find values of \(\theta\).
8. **Finding Maximum Area**:
- After solving for \(\theta\), substitute back into the area function to find the maximum area.
9. **Final Calculation**:
- The maximum area can be calculated and will yield:
\[
A_{\text{max}} = \frac{3\sqrt{3}}{8}
\]
### Conclusion:
The maximum value of the area of triangle PXY as X varies is \(\frac{3\sqrt{3}}{8}\).
