AMC12 2005 B

AMC12 2005 B · Q8

AMC12 2005 B · Q8. It mainly tests Functions basics, Graphs (coordinate plane).

For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$ ?

有多少个 $a$ 的取值使得直线 $y = x + a$ 经过抛物线 $y = x^2 + a^2$ 的顶点？

(A) 0 0

(B) 1 1

(D) 10 10

(E) infinitely many 无穷多个

Answer

Correct choice: (C)

正确答案：（C）

Solution

We see that the vertex of the quadratic function $y = x^2 + a^2$ is $(0,\,a^2)$. The y-intercept of the line $y = x + a$ is $(0,\,a)$. We want to find the values (if any) such that $a=a^2$. Solving for $a$, the only values that satisfy this are $0$ and $1$, so the answer is $\boxed{\mathrm{(C)}\ 2}$

二次函数 $y = x^2 + a^2$ 的顶点为 $(0,\,a^2)$。直线 $y = x + a$ 的 $y$ 轴截距为 $(0,\,a)$。我们要找（若存在）使得 $a=a^2$ 的 $a$ 值。解得满足条件的只有 $0$ 和 $1$，因此答案是 $\boxed{\mathrm{(C)}\ 2}$

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