5、(番禺 T9,)如图,抛物线y= ax2 + 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( ).
A. -1 B.2 C. -3 D. -2
视频信息
答案文本
视频字幕
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. We need to find the value of a c. Let's set up coordinates and analyze this step by step.
Let's establish a coordinate system. Since B is on the y-axis and O A B C forms a square, we can set up coordinates. Let the side length be s. Then O is at origin zero zero, A is at s zero, B is at s s, and C is at zero s. Notice that B is indeed on the y-axis when s equals the y-coordinate.
Now let's substitute the coordinates into the parabola equation y equals a x squared plus c. For point A at coordinates s comma zero, we get zero equals a s squared plus c. For point B at s comma s, we get s equals a s squared plus c. For point C at zero comma s, we get s equals a times zero squared plus c, which simplifies to s equals c. So we found that c equals s.
Now let's solve for parameter a. We know c equals s from the previous step. Substituting this into the equation from point A: zero equals a s squared plus c becomes zero equals a s squared plus s. Factoring out s, we get zero equals s times a s plus one. Since s is not zero for a valid square, we have a s plus one equals zero, which gives us a equals negative one over s.
Now let's calculate the final answer. We have a equals negative one over s and c equals s. Therefore, a times c equals negative one over s times s, which simplifies to negative s over s, which equals negative one. The answer is A, negative one.