过抛物线Y^2=4X的焦点的直线交抛物线于A,B两点,正三角形ABC的顶点C在该抛物线的准线上求ABC的边长
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过抛物线Y^2=4X的焦点的直线交抛物线于A,B两点,正三角形ABC的顶点C在该抛物线的准线上求ABC的边长
过抛物线Y^2=4X的焦点的直线交抛物线于A,B两点,正三角形ABC的顶点C在该抛物线的准线上求ABC的边长
过抛物线Y^2=4X的焦点的直线交抛物线于A,B两点,正三角形ABC的顶点C在该抛物线的准线上求ABC的边长
求出正三角形的AB边即可.
设A,B两点坐标(x1,y1),(x2,y2)
依据抛物线定义 ,A,B两点各自到准线的距离= 到焦点的距离
p =2,准线 x=-p/2 = -1
AB = x1-(-1) + x2-(-1) = x1+x2 +2
过抛物线焦点的直线为
y= k(x-p/2) = k(x-1)
k^2 (x-1)^2 = 4x
(x-1)^2 = 4x/k^2
x^2 -2x - 4/k^2 x +1 =0
所以
x1+x2 = 2+4/k^2
AB中点M(x3,y3)
x3=(x1+x2)/2 = 1+2/k^2
y3 = k(x3-1) = k+2/k -k= 2/k
过抛物线焦点的直线的垂线与准线交点N(x4,y4)
x4 =-1
(y4 -y3)/(x4-x3)= -1/k
y4-y3 = -1/k *(-1 -(1+2/k^2)) = 1/k *(2+2/k^2)
NM = AB sin60
NM^2 = AB^2 *3/4
(x4-x3)^2 + (y4-y3)^2 =(2+2k^2)^2 + 1/k^2 (2+2/k^2)^2
AB^2 *3/4 = (x1+x2 +2)^2 *3/4 = (4+4/k^2)^2 *3/4 = (2+2/k^2)^2 *3
所以
(2+2k^2)^2 + 1/k^2 (2+2/k^2)^2 = (2+2/k^2)^2 *3
1+ 1/k^2 = 3
AB = (4+4/k^2) = 4*(1+ 1/k^2) = 4*3 =12