What is the value of A_{1}?

Consider the following for the next two (02) items that follow : The circle x^{2}+y^{2}-2x=0 is partitioned by line y=x in two segments. Let A_{1}, A_{2} be the areas of major and minor segments respectively.

  1. A. \frac{\pi-2}{4}
  2. B. \frac{\pi+2}{4}
  3. C. \frac{3\pi-2}{4}
  4. D. \frac{3\pi+2}{4} ✓

Correct Answer: D. \frac{3\pi+2}{4}

Explanation

The circle is (x-1)^2+y^2=1, with center (1,0) and radius r=1. Its total area is \pi. The line y=x intersects it at 2x^2-2x=0 \implies x=0, 1, giving points (0,0) and (1,1). The chord length is \sqrt{1^2+1^2} = \sqrt{2}. The angle \theta subtended at the center by the chord satisfies \sin(\frac{\theta}{2}) = \frac{\sqrt{2}/2}{1} = \frac{1}{\sqrt{2}} \implies \theta = \frac{\pi}{2}. The area of the minor segment A_2 = \text{Area of sector} - \text{Area of triangle} = \frac{1}{4}\pi r^2 - \frac{1}{2}r^2\sin(\frac{\pi}{2}) = \frac{\pi}{4} - \frac{1}{2} = \frac{\pi-2}{4}. The major segment is A_1 = \pi - A_2 = \pi - \frac{\pi-2}{4} = \frac{3\pi+2}{4}.

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