已知直线x+y-1=0与椭圆x²/a²+y²/b²=1(a>b>0)交于A,B两点,M是线段AB上一点,向量AM=-向量BM,且M在直线L:y=1/2x上.(1)求椭圆的离心率.(2)若椭圆的焦点关于直线的对称点在单位圆
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![已知直线x+y-1=0与椭圆x²/a²+y²/b²=1(a>b>0)交于A,B两点,M是线段AB上一点,向量AM=-向量BM,且M在直线L:y=1/2x上.(1)求椭圆的离心率.(2)若椭圆的焦点关于直线的对称点在单位圆](/uploads/image/z/12507772-4-2.jpg?t=%E5%B7%B2%E7%9F%A5%E7%9B%B4%E7%BA%BFx%2By-1%3D0%E4%B8%8E%E6%A4%AD%E5%9C%86x%26%23178%3B%2Fa%26%23178%3B%2By%26%23178%3B%2Fb%26%23178%3B%3D1%EF%BC%88a%EF%BC%9Eb%EF%BC%9E0%EF%BC%89%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2CM%E6%98%AF%E7%BA%BF%E6%AE%B5AB%E4%B8%8A%E4%B8%80%E7%82%B9%2C%E5%90%91%E9%87%8FAM%3D-%E5%90%91%E9%87%8FBM%2C%E4%B8%94M%E5%9C%A8%E7%9B%B4%E7%BA%BFL%EF%BC%9Ay%3D1%2F2x%E4%B8%8A.%EF%BC%881%EF%BC%89%E6%B1%82%E6%A4%AD%E5%9C%86%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87.%EF%BC%882%EF%BC%89%E8%8B%A5%E6%A4%AD%E5%9C%86%E7%9A%84%E7%84%A6%E7%82%B9%E5%85%B3%E4%BA%8E%E7%9B%B4%E7%BA%BF%E7%9A%84%E5%AF%B9%E7%A7%B0%E7%82%B9%E5%9C%A8%E5%8D%95%E4%BD%8D%E5%9C%86)
已知直线x+y-1=0与椭圆x²/a²+y²/b²=1(a>b>0)交于A,B两点,M是线段AB上一点,向量AM=-向量BM,且M在直线L:y=1/2x上.(1)求椭圆的离心率.(2)若椭圆的焦点关于直线的对称点在单位圆
已知直线x+y-1=0与椭圆x²/a²+y²/b²=1(a>b>0)交于A,B两点,M是线段AB上一点,向量AM=-向量BM,且M在直线L:y=1/2x上.(1)求椭圆的离心率.(2)若椭圆的焦点关于直线的对称点在单位圆x²+y²=1上,求椭圆方程
已知直线x+y-1=0与椭圆x²/a²+y²/b²=1(a>b>0)交于A,B两点,M是线段AB上一点,向量AM=-向量BM,且M在直线L:y=1/2x上.(1)求椭圆的离心率.(2)若椭圆的焦点关于直线的对称点在单位圆
直线x+y-1=0
y=1-x代入x²/a²+y²/b²=1
b²x²+a²(1-x)²=a²b²
b²x²+a²x²-2a²x+a²-a²b²=0
(a²+b²)x²-2a²x+a²-a²b²=0
x1+x2=2a²/(a²+b²)
y1+y2=1-x1+1-x2=2-(x1+x2)=2-(2a²)/(a²+b²)=2b²/(a²+b²)
那么点M的坐标(a²/(a²+b²),b²/(a²+b²))
代入y=x/2
b²/(a²+b²)=a²/2(a²+b²)
a²=2b²
a²=b²+c²
a²=1/2a²+c²
c²=1/2a²
c²/a²=1/2
e²=1/2
e=√2/2