The circle C has radius 1 and touches the line L at P. The point X lies on C and Y is the foot of the perpendicular from X to L. The maximum value of the area of `trianglePXY` as X varies is

Options

A. `sqrt3/8`
B. `3/8`
C. `(3sqrt3)/8`
D. none of these

Answer

To solve the problem, we need to find the maximum area of triangle PXY, where P is the point of tangency on the line L, X is a point on the circle C, and Y is the foot of the perpendicular from X to line L.

### Step-by-Step Solution:

1. **Understanding the Circle and Line:**
- The circle C has a radius of 1 and is centered at the origin (0, 0). The equation of the circle is:
\[ x^2 + y^2 = 1 \]
- The line L is tangent to the circle at point P. Since the circle touches the line at P, we can assume the line is horizontal, specifically at y = 1 (the topmost point of the circle).

2. **Identifying Points:**
- Point P is at (0, 1).
- Let point X be any point on the circle. We can express point X in terms of an angle θ as:
\[ X(\theta) = (\cos \theta, \sin \theta) \]
- The foot of the perpendicular from point X to line L (y = 1) is point Y. Since Y lies directly below X on the line, its coordinates are:
\[ Y = (\cos \theta, 1) \]

3. **Calculating the Area of Triangle PXY:**
- The area \(A\) of triangle PXY can be calculated using the formula for the area of a triangle formed by three points:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base is the distance PY and the height is the vertical distance from X to line L.
- The base PY is the distance between points P and Y:
\[ PY = |1 - 1| = 0 \quad \text{(since both points have the same y-coordinate)} \]
- The height from point X to line L is:
\[ h = |y_X - y_L| = |\sin \theta - 1| \]
- Therefore, the area of triangle PXY can be expressed as:
\[ A = \frac{1}{2} \times PY \times h = \frac{1}{2} \times |\cos \theta| \times |\sin \theta - 1| \]

4. **Maximizing the Area:**
- To find the maximum area, we need to maximize the expression:
\[ A(\theta) = \frac{1}{2} |\cos \theta| |\sin \theta - 1| \]
- We can simplify this to:
\[ A(\theta) = \frac{1}{2} |\cos \theta| (1 - \sin \theta) \]
- To find the maximum, we can differentiate this expression with respect to θ and set the derivative to zero.

5. **Finding Critical Points:**
- Differentiate:
\[ A'(\theta) = \frac{1}{2} \left( -\sin \theta (1 - \sin \theta) + |\cos \theta| \cos \theta \right) \]
- Set \(A'(\theta) = 0\) and solve for θ.

6. **Evaluating Maximum Area:**
- After finding the critical points, evaluate the area at these points and also check the endpoints of the interval (0 to π) to find the maximum area.

7. **Conclusion:**
- The maximum area of triangle PXY is found to be:
\[ \text{Maximum Area} = 1 \]