2017 AMC 12B Problems/Problem 6

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Problem 6

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

Solution

Solution 1 Because the two points are a diameter the center must be half way between them at the point (4,3). The distance from (0,0) to (4,3) is 5 therefore the circle has a radius of 5. The equation of the circle can thus be written as (x-4)^2+(y-3)^2=25. To find the x intercept y must be 0, so (x-4)^2+(0-3)^2=25. Then (x-4)^2=16. x-4=4. x=8

See Also

2017 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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All AMC 12 Problems and Solutions

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