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Two circles of radii a and 1 (where a < 1) touch each other externally and are both inscribed in the region bounded by the lower semicircle y = -sqrt(4 - x²) and the x-axis (i.e., inside the lower half of the circle x² + y² = 4). Which of the following is/are correct?

1. 100*(a + 1) = 125
2. The number of such pairs of circles is 2
3. a = 1/2
4. The points of tangency of the circle of radius a with y = -sqrt(4 - x²) are (plus or minus 4*sqrt(2)/3, -2/3)

Correct answer: a = 1/2

Solution

Using the inscribed circle conditions and external tangency, we get: for radius 1: x₁² = 4-4*1 = 0, so x₁=0 (on y-axis). For radius a: xₐ² = 4-4a. External tangency: xₐ² + (1-a)² = (a+1)² => 4-4a + 1-2a+a² = a²+2a+1 => 4-4a = 4a => 8a=4 => a=1/2.

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