AMC10 2014 A
AMC10 2014 A · Q14
AMC10 2014 A · Q14. It mainly tests Graphs (coordinate plane), Coordinate geometry.
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6, 8)$ have a sum of zero. What is the area of $\triangle APQ$?
两条相互垂直的直线相交于点$A(6, 8)$，其$y$轴截距$P$和$Q$之和为零。$\triangle APQ$的面积是多少？
(A)
45
45
(B)
48
48
(C)
54
54
(D)
60
60
(E)
72
72
Answer
Correct choice: (D)
正确答案：（D）
Solution
Answer (D): Let the $y$-intercepts of lines $PA$ and $QA$ be $\pm b$. Then their slopes are $\frac{8\pm b}{6}$. Setting the product of the slopes to $-1$ and solving yields $b=\pm 10$. Therefore $\triangle APQ$ has base $20$ and altitude $6$, for an area of $60$.
答案（D）：设直线 $PA$ 和 $QA$ 的 $y$ 轴截距为 $\pm b$。则它们的斜率为 $\frac{8\pm b}{6}$。令两斜率的乘积为 $-1$ 并求解，得 $b=\pm 10$。因此，$\triangle APQ$ 的底为 $20$、高为 $6$，面积为 $60$。
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