AMC12 2017 B

AMC12 2017 B · Q6

AMC12 2017 B · Q6. It mainly tests Circle theorems, Coordinate geometry.

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?

以点 (0, 0) 和 (8, 6) 为直径端点的圆与 x 轴相交于另一个点。这个点的 x 坐标是多少？

(A) $4\sqrt{2}$ $4\sqrt{2}$

(B) 6 6

(D) 8 8

(E) $6\sqrt{2}$ $6\sqrt{2}$

Answer

Correct choice: (D)

正确答案：（D）

Solution

The center of the circle is the midpoint of the diameter, which is (4, 3), and the radius is $\sqrt{4^{2} + 3^{2}} = 5$. Therefore the equation of the circle is $(x - 4)^{2} + (y - 3)^{2} = 25$. If $y = 0$, then $(x - 4)^{2} = 16$, so $x = 0$ or $x = 8$. The circle intersects the x-axis at (8, 0).

圆的圆心是直径的中点，为 (4, 3)，半径为 $\sqrt{4^{2} + 3^{2}} = 5$。因此圆的方程为 $(x - 4)^{2} + (y - 3)^{2} = 25$。当 $y = 0$ 时，$(x - 4)^{2} = 16$，所以 $x = 0$ 或 $x = 8$。圆与 x 轴相交于 (8, 0)。

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