2017 AMC 12B Problems/Problem 6

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?

$\textbf{(A) } 4\sqrt{2} \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 5\sqrt{2} \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

Solution

Because the two points are on a diameter, the center must be halfway between them at the point (4,3). The distance from (0,0) to (4,3) is 5 so the circle has radius 5. Thus, the equation of the circle is $(x-4)^2+(y-3)^2=25$ .

To find the x-intercept, y must be 0, so $(x-4)^2+(0-3)^2=25$ , so $(x-4)^2=16$ , $x-4=4$ , $x=8, \boxed{D}$ .

Written by: SilverLion

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2017 AMC 12B Problems/Problem 6

Problem 6

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