AMC12 2021 B

AMC12 2021 B · Q5

AMC12 2021 B · Q5. It mainly tests Coordinate geometry, Transformations.

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b - a ?$

平面直角坐标系中，点 $P(a,b)$ 先绕点 $(1,5)$ 逆时针旋转 $90^\circ$，然后关于直线 $y = -x$ 作反射。经过这两个变换后，$P$ 的像位于 $(-6,3)$。求 $b - a$ 的值。

(A) 1 1

(B) 3 3

(D) 7 7

(E) 9 9

Answer

Correct choice: (D)

正确答案：（D）

Solution

The final image of $P$ is $(-6,3)$. We know the reflection rule for reflecting over $y=-x$ is $(x,y) \rightarrow (-y, -x)$. So before the reflection and after rotation the point is $(-3,6)$. By definition of rotation, the slope between $(-3,6)$ and $(1,5)$ must be perpendicular to the slope between $(a,b)$ and $(1,5)$. The first slope is $\frac{5-6}{1-(-3)} = \frac{-1}{4}$. This means the slope of $P$ and $(1,5)$ is $4$. Rotations also preserve distance to the center of rotation, and since we only "travelled" up and down by the slope once to get from $(-3,6)$ to $(1,5)$ it follows we shall only use the slope once to travel from $(1,5)$ to $P$. Therefore point $P$ is located at $(1+1, 5+4) = (2,9)$. The answer is $9-2 = 7 = \boxed{\textbf{(D)} ~7}$.

$P$ 的最终像是 $(-6,3)$。关于 $y=-x$ 反射的规则是 $(x,y) \rightarrow (-y, -x)$。因此，反射前旋转后的点是 $(-3,6)$。根据旋转定义，点 $(-3,6)$ 与 $(1,5)$ 连线的斜率必须与 $(a,b)$ 与 $(1,5)$ 连线的斜率垂直。第一条连线的斜率为 $\frac{5-6}{1-(-3)} = \frac{-1}{4}$。因此，$P$ 与 $(1,5)$ 的连线斜率为 $4$。旋转还保持到旋转中心的距离，由于从 $(-3,6)$ 到 $(1,5)$ 只沿斜率方向“移动”一次，因此从 $(1,5)$ 到 $P$ 也只需沿斜率移动一次。因此点 $P$ 位于 $(1+1, 5+4) = (2,9)$。答案是 $9-2 = 7 = \boxed{\textbf{(D)} ~7}$。

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