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On an engineering grid, a circular sensor zone is drawn so that the points A(1, 2) and B(9, 8) are the two endpoints of a diameter of the zone's boundary circle. A repair access point sits on the boundary circle at height y = 1, on the side of the circle to the right of the center. What is the x-coordinate of the repair access point?
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Correct answer
C
The two given points are the ends of a diameter, so the circle's center is their midpoint and the radius is half the distance between them.
Center: the midpoint of A(1, 2) and B(9, 8) is ((1 + 9)/2, (2 + 8)/2) = (5, 5). Radius: the distance from A to B is √((9 − 1)² + (8 − 2)²) = √(64 + 36) = √100 = 10, so the radius is 10 ÷ 2 = 5. The boundary circle is (x − 5)² + (y − 5)² = 25.
Set y = 1 to find the point on the circle at that height: (x − 5)² + (1 − 5)² = 25, so (x − 5)² + 16 = 25, giving (x − 5)² = 9 and x − 5 = ±3. The two solutions are x = 8 and x = 2. The access point sits to the right of the center, so x = 8. The answer is (C).
Quick check: the point at y = 1 is 4 units below the center, and the radius is 5, so the horizontal distance from the center to the boundary at that height is √(5² − 4²) = √9 = 3. To the right of the center, that lands at 5 + 3 = 8, confirming (C).
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