椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点分别为F1,F2,过F1的直线l与椭圆相交于A,B两点,若∠AF1F2=60且向量AF1*向量AF2=0,则椭圆的离心率为?
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![椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点分别为F1,F2,过F1的直线l与椭圆相交于A,B两点,若∠AF1F2=60且向量AF1*向量AF2=0,则椭圆的离心率为?](/uploads/image/z/5451002-26-2.jpg?t=%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%EF%BC%88a%3Eb%3E0%29%E7%9A%84%E5%B7%A6%E3%80%81%E5%8F%B3%E7%84%A6%E7%82%B9%E5%88%86%E5%88%AB%E4%B8%BAF1%2CF2%2C%E8%BF%87F1%E7%9A%84%E7%9B%B4%E7%BA%BFl%E4%B8%8E%E6%A4%AD%E5%9C%86%E7%9B%B8%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E8%8B%A5%E2%88%A0AF1F2%3D60%E4%B8%94%E5%90%91%E9%87%8FAF1%2A%E5%90%91%E9%87%8FAF2%3D0%2C%E5%88%99%E6%A4%AD%E5%9C%86%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA%3F)
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椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点分别为F1,F2,过F1的直线l与椭圆相交于A,B两点,若∠AF1F2=60且向量AF1*向量AF2=0,则椭圆的离心率为?
椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点分别为F1,F2,过F1的直线l与椭圆相交于A,B两点,若∠AF1F2=60
且向量AF1*向量AF2=0,则椭圆的离心率为?
椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左、右焦点分别为F1,F2,过F1的直线l与椭圆相交于A,B两点,若∠AF1F2=60且向量AF1*向量AF2=0,则椭圆的离心率为?
向量AF1*向量AF2=0,则向量AF1垂直AF2,
〈F1AF2=90度,
|F1F2|=2c,
|AF2|/|F1F2|=sin60°=√3/2,(1)
|AF1|/|F1F2|=cos60°=1/2,(2)
(1)+(2)式,
(|AF1|+|AF2|)/|F1F2|=(√3+1)/2,
|AF1|+|AF2|=2a,
2a/2c=(√3+1)/2,
c/a=2/(√3+1)=√3-1,
离心率e=c/a=√3-1.