已知,抛物线y=x²-4x+3与x轴交于A,B(A点在B点左侧)与y轴交于C点.将此抛物线向左平移,平移后的抛物线交直线AC于M,N两点,请求出使得MN=2AC的抛物线的解析式
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![已知,抛物线y=x²-4x+3与x轴交于A,B(A点在B点左侧)与y轴交于C点.将此抛物线向左平移,平移后的抛物线交直线AC于M,N两点,请求出使得MN=2AC的抛物线的解析式](/uploads/image/z/8605358-62-8.jpg?t=%E5%B7%B2%E7%9F%A5%2C%E6%8A%9B%E7%89%A9%E7%BA%BFy%3Dx%26%23178%3B-4x%2B3%E4%B8%8Ex%E8%BD%B4%E4%BA%A4%E4%BA%8EA%2CB%EF%BC%88A%E7%82%B9%E5%9C%A8B%E7%82%B9%E5%B7%A6%E4%BE%A7%EF%BC%89%E4%B8%8Ey%E8%BD%B4%E4%BA%A4%E4%BA%8EC%E7%82%B9.%E5%B0%86%E6%AD%A4%E6%8A%9B%E7%89%A9%E7%BA%BF%E5%90%91%E5%B7%A6%E5%B9%B3%E7%A7%BB%2C%E5%B9%B3%E7%A7%BB%E5%90%8E%E7%9A%84%E6%8A%9B%E7%89%A9%E7%BA%BF%E4%BA%A4%E7%9B%B4%E7%BA%BFAC%E4%BA%8EM%2CN%E4%B8%A4%E7%82%B9%2C%E8%AF%B7%E6%B1%82%E5%87%BA%E4%BD%BF%E5%BE%97MN%3D2AC%E7%9A%84%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%9A%84%E8%A7%A3%E6%9E%90%E5%BC%8F)
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已知,抛物线y=x²-4x+3与x轴交于A,B(A点在B点左侧)与y轴交于C点.将此抛物线向左平移,平移后的抛物线交直线AC于M,N两点,请求出使得MN=2AC的抛物线的解析式
已知,抛物线y=x²-4x+3与x轴交于A,B(A点在B点左侧)与y轴交于C点.将此抛物线向左平移
,平移后的抛物线交直线AC于M,N两点,请求出使得MN=2AC的抛物线的解析式
已知,抛物线y=x²-4x+3与x轴交于A,B(A点在B点左侧)与y轴交于C点.将此抛物线向左平移,平移后的抛物线交直线AC于M,N两点,请求出使得MN=2AC的抛物线的解析式
y = (x - 1)(x - 3)
A(1,0),B(3,0),C(0,3)
AC = √10,MN = 2√10
向左平移后p后的抛物线:y = (x - 1+ p)(x - 3+p)
AC的解析式:x + y/3 = 1,y = -3(x - 1)
联立得:x² + (2p - 1)x + p² - 4p = 0
x₁ + x₂ = 1 - 2p
x₁x₂ = p² - 4p
MN² = 40 = (x₁ - x₂)²+ (y₁ - y₂)² = (x₁ - x₂)²+ (-3x₁ +3 + 3x₂- 3)² = 10(x₁ - x₂)²
(x₁ - x₂)² = 4 = (x₁ + x₂)² - 4x₁x₂ = 4p² - 4p + 1 - 4(p² - 4p) = 12p + 1
p = 1/4
向左平移后1/4后的抛物线:y = (x - 1+ 1/4)(x - 3+1/4) = (x - 3/4)(x - 11/4)