向量AB (2,4)的起点A和终点B都在椭圆4x平方+15y平方=64上,求A和B的坐标

题目

向量AB (2,4)的起点A和终点B都在椭圆4x平方+15y平方=64上,求A和B的坐标
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答案

let A(x1,y1),B(x2,y2)
AB = (2,4)
let the line pass through AB be
L:y=2x+C (1)
E:4x^2+15y^2=64 (2)
Sub (1) into (2)
4x^2 + 15(2x+C)^2=64
64x^2+60Cx+ 15C^2-64=0
x1+x2 = -60C/64
x1x2=(15C^2-64)/64
(x2-x1)^2 = (x2+x1)^2 -4x1x2
4= (-60C/64)^2-4(15C^2-64)/64)
16384 =3600C^2-11520C^2+16384
C=0
L:y=2x (3)
sub(3) into (2)
4x^2+15y^2=64
4x^2+60x^2 =64
x=1 or -1
x=1,y=2
x=-1,y=-2
ie B(1,2),A(-1,-2)