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Consider the region R = {(x, y): x > 0 and y² <= 4 - x}. Among all circles that lie entirely inside R with centres on the x-axis, let C be the one of largest radius. If C meets the boundary curve y² = 4 - x at the point (alpha, beta), find the radius of C and the value of alpha.

1. radius = sqrt(15)/2, alpha = 3/2
2. radius = sqrt(15)/4, alpha = 1/4
3. radius = sqrt(15)/2, alpha = 1/2
4. radius = 2, alpha = 0

Correct answer: radius = sqrt(15)/2, alpha = 3/2

Solution

This is the JEE Advanced 2021 region problem; the largest inscribed circle on the x-axis has radius sqrt(15)/2 and meets y² = 4 - x at alpha = 3/2.

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