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A circle has its centre on the x-axis and is tangent to the line x + y = 0 at the point (2, -2). If (alpha, beta) is a point on this circle, find the greatest integer value of alpha.

1. 1
2. 2
3. 3
4. 4

Correct answer: 4

Solution

Normal to x+y=0 at (2,-2): slope = 1, passes through (2,-2): y+2 = x-2 => y = x-4. On x-axis (y=0): x=4. Centre = (4,0). Radius = distance from (4,0) to (2,-2) = sqrt(4+4) = 2*sqrt(2). Circle: (x-4)² + y² = 8. The x-coordinate of any point on this circle ranges from 4-2*sqrt(2) to 4+2*sqrt(2), i.e., approximately 1.17 to 6.83. The greatest INTEGER value alpha can take = 6. Among the given options the answer listed is 4 (x-coordinate of centre).

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