AMC10 2017 B

AMC10 2017 B · Q10

AMC10 2017 B · Q10. It mainly tests Linear equations, Coordinate geometry.

The lines with equations $ax - 2y = c$ and $2x + by = -c$ are perpendicular and intersect at $(1, -5)$. What is $c$?

方程 $ax - 2y = c$ 和 $2x + by = -c$ 表示两条垂直且相交于点 $(1, -5)$ 的直线。$c$ 的值是多少？

(A) -13 -13

(B) -8 -8

(D) 8 8

(E) 13 13

Answer

Correct choice: (E)

正确答案：（E）

Solution

Because the lines are perpendicular, their slopes, a/2 and −2/b, are negative reciprocals, so a = b. Substituting b for a and using the point (1, −5) yields the equations b + 10 = c and 2 − 5b = −c. Adding the two equations yields 12 − 4b = 0, so b = 3. Thus c = 3 + 10 = 13.

因为两条直线垂直，它们的斜率 $\frac{a}{2}$ 和 $\frac{-2}{b}$ 是负互为倒数，故 $a=b$。代入 $b$ 代替 $a$，并利用点 $(1,-5)$，得到方程 $b+10=c$ 和 $2-5b=-c$。将两式相加得 $12-4b=0$，故 $b=3$。从而 $c=3+10=13$。

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