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2017 AMC 12B Problem 6

Problem 6 of 25EasierGeometry

The circle having (0,0)(0, 0) and (8,6)(8, 6) as the endpoints of a diameter intersects the xx-axis at a second point. What is the xx-coordinate of this point?

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Solution

The center is the midpoint of the diameter, (4,3),(4, 3), and the radius is 42+32=5.\sqrt{4^2 + 3^2} = 5. The circle is (x−4)2+(y−3)2=25.(x - 4)^2 + (y - 3)^2 = 25. Setting y=0y = 0 gives (x−4)2=16,(x - 4)^2 = 16, so x=0x = 0 or x=8.x = 8. The second intersection with the xx-axis is at x=8.x = 8. Thus, the correct answer is D.

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Concepts: coordinate geometry · circle · midpoint

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.