AMC10 2015 B

AMC10 2015 B · Q12

AMC10 2015 B · Q12. It mainly tests Coordinate geometry, Counting in geometry (lattice points).

For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius 10 centered at (5,5)?

有整数$x$使得点$(x, -x)$在以(5,5)为圆心、半径为10的圆内或圆周上？

(A) 11 11

(B) 12 12

(D) 14 14

(E) 15 15

Answer

Correct choice: (A)

正确答案：（A）

Solution

The circle intersects the line $y = -x$ at the points $A = (-5, 5)$ and $B = (5, -5)$. Segment $\overline{AB}$ is a chord of the circle and contains 11 points with integer coordinates.

圆与直线$y = -x$相交于点$A = (-5, 5)$和$B = (5, -5)$。线段$\overline{AB}$是圆的一个弦，包含11个整数坐标点。

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