Question

对多少个 $a$ 的值，直线 $y = x + a$ 经过抛物线 $y = x^2 + a^2$ 的顶点？

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

$0$

$1$

$2$

$10$

无限多个

infinitely many

$y = x^2 + a^2$ 的顶点是 $(0, a^2)$ 。

The vertex of $y = x^2 + a^2$ is $(0, a^2)$

把顶点代入直线，得到 $a^2 = a$ 。

Substitute the vertex into the line to get $a^2 = a$

抛物线 $y = x^2 + a^2$ 的顶点为 $(0, a^2)$ 。

直线 $y = x + a$ 经过该点当且仅当 $a^2 = 0 + a$ ，即 $a^2 - a = 0$ 。

所以 $a = 0$ 或 $a = 1$ ，共有 $2$ 个值。

所以正确答案是 C。

The vertex of the parabola $y = x^2 + a^2$ is $(0, a^2).$

The line $y = x + a$ passes through it exactly when $a^2 = 0 + a,$ that is $a^2 - a = 0.$

This gives $a = 0$ or $a = 1,$ so there are $2$ values.

Thus, the correct answer is C.