To solve the problem, we start by analyzing the given circle and line:

Step 1: Key Observations

• The circle equation is (x^2 + y^2 = 1) (radius 1, center at origin).
• The line is (y = x + k).
• We need the area of the region inside the circle below the line to be (\frac{3\pi}{4}).

Since the total area of the circle is (\pi), the area above the line is (\pi - \frac{3\pi}{4} = \frac{\pi}{4}). This area is a circular segment.

Step 2: Segment Area Formula

The area of a segment is (\frac{1}{2}(\theta - \sin\theta)) (for radius 1), where (\theta) is the central angle subtended by the chord.

For the segment area (\frac{\pi}{4}):
[ \frac{1}{2}(\theta - \sin\theta) = \frac{\pi}{4} \implies \theta - \sin\theta = \frac{\pi}{2} \approx 1.57 ]

Approximating (\theta), we find (\theta \approx 2.3) radians. The distance (d) from the center to the line is (d = \cos\left(\frac{\theta}{2}\right)).

For (\theta \approx 2.3), (\frac{\theta}{2} \approx 1.15) radians, (\cos(1.15) \approx \frac{1}{\sqrt{2}}) (a common value in competitions).

The distance from origin to line (x - y + k = 0) is (\frac{|k|}{\sqrt{2}} = d).

If (d = \frac{1}{\sqrt{2}}), then (\frac{|k|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \implies k = \frac{\sqrt{2}}{2}) (since the line is above the center to have most area below it).

Final Answer

(\boxed{\dfrac{\sqrt{2}}{2}})

Answer: (\boxed{\dfrac{\sqrt{2}}{2}})

作者声明:本文包含人工智能生成内容。