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Two circles of radii a and b touch each other externally and are both inscribed in the region bounded by the semicircle y = sqrt(1 - x²) and the x-axis. If b = 1/2, find the value of 4a.

1. 1
2. 2/3
3. 3/2
4. 5/3

Correct answer: 2/3

Solution

Using the geometric constraints for both circles touching the x-axis, internally tangent to the unit semicircle, and externally tangent to each other, we solve for a. With b = 1/2 the larger circle center is at (sqrt(3)/2, 1/2). Setting up the tangency condition for the smaller circle gives a = 1/6, so 4a = 2/3.

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