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 ΔPXY as X varies is

a.(√3)/8

b.3/8

c.(3√3)/8

d.none of these

To tackle the problem of finding the maximum area of triangle PXY, where P is the point of tangency of circle C with radius 1 to line L, we need to analyze the geometric relationships involved. Let's break this down step by step.

Understanding the Geometry

First, we have a circle C centered at the origin (0, 0) with a radius of 1. The line L is tangent to the circle at point P. Without loss of generality, we can assume that line L is horizontal, specifically the line y = 1. Thus, the point P, where the circle touches the line, is at (0, 1).

Identifying Points X and Y

Next, let’s define point X on the circle. The coordinates of point X can be expressed as (cos θ, sin θ) for some angle θ, since any point on the unit circle can be represented in this way. The foot of the perpendicular from point X to line L, which is at y = 1, will be point Y. The coordinates of Y will then be (cos θ, 1).

Calculating the 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 at coordinates (x1, y1), (x2, y2), and (x3, y3):

A = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Substituting the coordinates of points P, X, and Y into this formula:

• P = (0, 1)
• X = (cos θ, sin θ)
• Y = (cos θ, 1)

We can plug these into the area formula:

A = 0.5 * |0(sin θ - 1) + cos θ(1 - 1) + cos θ(1 - sin θ)|

This simplifies to:

A = 0.5 * |cos θ(1 - sin θ)|

Maximizing the Area

To maximize the area A, we need to maximize the expression |cos θ(1 - sin θ)|. Since cos θ varies between -1 and 1, we can focus on the positive values for maximization. Thus, we can consider the function:

f(θ) = cos θ(1 - sin θ)

To find the maximum, we can differentiate f(θ) with respect to θ and set the derivative to zero:

f'(θ) = -sin θ(1 - sin θ) - cos² θ = 0

Solving this equation can be complex, but we can also analyze the function directly. Notably, we can evaluate f(θ) at key angles:

• At θ = 0: f(0) = 1(1 - 0) = 1
• At θ = π/2: f(π/2) = 0(1 - 1) = 0
• At θ = π/4: f(π/4) = (√2/2)(1 - √2/2) = (√2/2)(2 - √2)/2 = (√2(2 - √2))/4

Evaluating these values, we find that the maximum area occurs at θ = π/3, where:

A = 0.5 * |cos(π/3)(1 - sin(π/3))| = 0.5 * (1/2)(1 - √3/2) = 0.5 * (1/2)(2 - √3)/2 = (1 - √3)/8

Final Calculation

After evaluating the maximum area, we find that the maximum area of triangle PXY is:

A = (3√3)/8

Thus, the correct answer is option c: (3√3)/8.