Multiple choice

Find the area of the portion of the circle $\displaystyle { x }^{ 2 }+{ \left( y-2 \right) }^{ 2 }=4$ in Quadrant 1.

  1. 0
  2. $\displaystyle \pi $
  3. $\displaystyle 2\pi $
  4. $\displaystyle 4\pi $
  5. $\displaystyle 16\pi $
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C Correct answer
Explanation

The circle is x^2 + (y-2)^2 = 4, centered at (0,2) with radius 2. It is tangent to the x-axis at (0,0). The portion in Quadrant 1 is the right half of the circle, which is a semicircle. Area = (1/2) * pi * r^2 = (1/2) * pi * 4 = 2pi.

AI explanation

The equation of the circle is x^2 + (y-2)^2 = 4, which means it has a center at (0, 2) and a radius of 2. Because the center lies on the y-axis, the circle is perfectly divided into two equal semicircles by the x-axis, with exactly half of the circle lying in Quadrant 1. The total area of the circle is pi * r^2 = 4*pi, so the area of the portion in Quadrant 1 is exactly half of this, which is 2*pi.

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