讲解这个题目---Question:
5. (番禺T9)如图,抛物线y=ax^2+c 经过正方形OABC的三个顶点A,B,C,点B在y轴上,则 ac 的值为( ).
Chart/Diagram Description:
* Type: Coordinate plane with a parabola and a square.
* Coordinate Axes: X-axis and Y-axis intersecting at the origin O. The Y-axis is vertical, pointing upwards, labeled 'y'. The X-axis is horizontal, pointing to the right, labeled 'x'. The origin is labeled 'O'.
* Geometric Shape: A square labeled OABC. Vertex O is at the origin (0,0). Vertex B is on the positive Y-axis. Vertices A and C are located in the first and second quadrants, respectively.
* Parabola: A downward-opening parabola passes through points A, B, and C.
* Points:
* O: Origin (0,0).
* A: A vertex of the square in the first quadrant, on the parabola.
* B: A vertex of the square on the positive Y-axis, also the vertex of the parabola.
* C: A vertex of the square in the second quadrant, on the parabola.
* Labels and Annotations: The diagram is labeled "(第9题)".
Options:
A. -1
B. 2
C. -3
D. -2
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答案文本
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Let's analyze this problem step by step. We have a parabola y equals a x squared plus c that passes through three vertices of square O A B C. Point B is on the y-axis, and we need to find the value of a c.
我们来分析这道关于抛物线和正方形的题目。抛物线 y 等于 a x 的平方加 c 经过正方形 O A B C 的三个顶点 A、B、C，其中点 B 在 y 轴上。我们需要求出 a c 的值。
现在让我们确定关键观察。由于 B 在 y 轴上，而抛物线 y 等于 a x 的平方加 c 关于 y 轴对称，所以点 B 必须是抛物线的顶点。因此，B 的坐标为 (0, c)。正方形 O A B C 的 O 在原点。
现在我们确定正方形的各个顶点坐标。O在原点(0, 0)，B在y轴上坐标为(0, c)。由于OABC是正方形，边长为c，我们可以确定A的坐标为(c, c)，C的坐标为(-c, c)。
现在我们建立方程组。将各个顶点的坐标代入抛物线方程 y 等于 a x 的平方加 c。对于点 A(c, c)，我们得到 c 等于 a c 的平方加 c，化简得到 0 等于 a c 的平方。对于点 C(-c, c)，同样得到 0 等于 a c 的平方。
最后我们求解答案。经过仔细分析，我们发现正方形的边长应该使得A和C两点都在抛物线上。设c等于2，通过计算可得a等于负1，因此ac的值等于负1乘以2，等于负2。答案是D选项。
Now let's determine the coordinates of the square vertices. We know O is at the origin and B is at (0, c). For square OABC, the diagonals OB and AC are perpendicular and bisect each other. This gives us A at (c/2, c/2) and C at (-c/2, c/2).
Now we substitute the coordinates into the parabola equation. For point A at (c/2, c/2), we substitute into y equals a x squared plus c. This gives us c/2 equals a times (c/2) squared plus c. Simplifying step by step, we get negative c/2 equals a times c squared over 4. Multiplying both sides by 4 and dividing by c, we find that ac equals negative 2.
We have found our final answer. From our calculation, ac equals negative 2. Looking at the multiple choice options: A is negative 1, B is 2, C is negative 3, and D is negative 2. Therefore, the correct answer is option D, negative 2.