运用点差法,求弦中点的轨迹方程.已知抛物线y^2=6x,求过点P(0,1)的直线被抛物线所截得弦的中点的轨迹方程.
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![运用点差法,求弦中点的轨迹方程.已知抛物线y^2=6x,求过点P(0,1)的直线被抛物线所截得弦的中点的轨迹方程.](/uploads/image/z/12075159-39-9.jpg?t=%E8%BF%90%E7%94%A8%E7%82%B9%E5%B7%AE%E6%B3%95%2C%E6%B1%82%E5%BC%A6%E4%B8%AD%E7%82%B9%E7%9A%84%E8%BD%A8%E8%BF%B9%E6%96%B9%E7%A8%8B.%E5%B7%B2%E7%9F%A5%E6%8A%9B%E7%89%A9%E7%BA%BFy%5E2%3D6x%2C%E6%B1%82%E8%BF%87%E7%82%B9P%EF%BC%880%2C1%EF%BC%89%E7%9A%84%E7%9B%B4%E7%BA%BF%E8%A2%AB%E6%8A%9B%E7%89%A9%E7%BA%BF%E6%89%80%E6%88%AA%E5%BE%97%E5%BC%A6%E7%9A%84%E4%B8%AD%E7%82%B9%E7%9A%84%E8%BD%A8%E8%BF%B9%E6%96%B9%E7%A8%8B.)
运用点差法,求弦中点的轨迹方程.已知抛物线y^2=6x,求过点P(0,1)的直线被抛物线所截得弦的中点的轨迹方程.
运用点差法,求弦中点的轨迹方程.
已知抛物线y^2=6x,求过点P(0,1)的直线被抛物线所截得弦的中点的轨迹方程.
运用点差法,求弦中点的轨迹方程.已知抛物线y^2=6x,求过点P(0,1)的直线被抛物线所截得弦的中点的轨迹方程.
设两交点为: A(a²/6, a), B(b²/6, b),
AB的方程: (y - b)/(a - b) = (x - b²/6)/(a²/6 - b²/6)
y - b = (6x - b²)/(a + b)
P(0,1)在AB上: 1 - b = -b²/(a + b)
a + b = ab (i)
设中点M(x, y):
x= (a² + b²)/12, a² + b² = 12x (ii)
y = (a+ b)/2, a + b = 2y (iii)
(ii)可以变为: 12x = a² + b² = (a + b)² - 2ab = (a + b)² - 2(a + b) (利用(i))
= (2y)² - 2*2y
= 4y² - 4y
y² - y = 3x
(y - 1/2)² = 3(x + 1/12)
也是抛物线, 它是y² = 3x向上平移1/2, 向左平移1/12得到的;
y² - y = 3x与y² = 6x交于O和A(2/3, 2), 在0 < y < 2时, y² - y = 3x的图象在y² = 6x的图象之外,应当排除.
答案: 轨迹方是y² - y = 3x, y < 0或y > 2