椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左焦点为F1(-c,0),A(-a,0)B(0,b)是两个顶点如果F1到直线AB的距离为b/√7,求椭圆的离心率e
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![椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左焦点为F1(-c,0),A(-a,0)B(0,b)是两个顶点如果F1到直线AB的距离为b/√7,求椭圆的离心率e](/uploads/image/z/2490689-65-9.jpg?t=%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%E7%9A%84%E5%B7%A6%E7%84%A6%E7%82%B9%E4%B8%BAF1%28-c%2C0%29%2CA%28-a%2C0%29B%280%2Cb%29%E6%98%AF%E4%B8%A4%E4%B8%AA%E9%A1%B6%E7%82%B9%E5%A6%82%E6%9E%9CF1%E5%88%B0%E7%9B%B4%E7%BA%BFAB%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BAb%2F%E2%88%9A7%2C%E6%B1%82%E6%A4%AD%E5%9C%86%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87e)
椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左焦点为F1(-c,0),A(-a,0)B(0,b)是两个顶点如果F1到直线AB的距离为b/√7,求椭圆的离心率e
椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左焦点为F1(-c,0),A(-a,0)B(0,b)是两个顶点
如果F1到直线AB的距离为b/√7,求椭圆的离心率e
椭圆x^2/a^2+y^2/b^2=1(a>b>0)的左焦点为F1(-c,0),A(-a,0)B(0,b)是两个顶点如果F1到直线AB的距离为b/√7,求椭圆的离心率e
AB的方程为:y=bx/a+b,即:bx-ay+ab=0
由点到直线的距离公式,F1(-c,0)到AB的距离d=|-bc+ab|/√(a²+b²)=b(a-c)/√(a²+b²)=b/√7,
则:7(a-c)²=a²+b²
7a²-14ac+7c²=a²+a²-c²
8c²-14ac+5a²=0
(2c-a)(4c-5a)=0
得:a=2c,或5a=4c(舍去,因为椭圆中:a>c)
所以,a=2c
则离心率e=c/a=1/2