如图,已知抛物线y=ax2+bx+c经过O(0,0)
1个回答
(1)
经过O,A(4,0),可表达为y = ax(x - 4)
经过B(3,√3):-3a = √3
a = -√3/3,b = 4√3/3
抛物线的函数解析式:y = -√3/3(x² - 4x)
(2)
t秒时:P(t,0)
(i) Q在AB上
AB的解析式:(y - 0)/(√3 - 0) = (x - 4)/(3 - 4),y = -√3(x - 4)
AB = √[(3 - 4)² + (√3 - 0)²] = 2
0 < t < 2
设Q(q,-√3(q - 4)),显然0 < q < 4
AQ² = OP² = t² = (q - 4)² + [-√3(q - 4) - 0]² = 4(q - 4)²
q = 4 - t/2 (舍去q = 4 + t/2)
Q(4 - t/2,√3t/2)
S = (1/2)PA*Q的纵坐标 = (1/2)(4 - t)*√3t/2 = √3(4 - t)t/4
(ii)Q在BC上
2 < t < 4
QB = t - AB = t - 2,Q的横坐标 = B的横坐标 - QB = 3 - (t - 2) = 5 - t
Q(5 - t,√3)
S = (1/2)PA*Q的纵坐标 = (1/2)(4 - t)√3 = √3(4 - t)/2
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