已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF···已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF,则此椭圆的离心率为?
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![已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF···已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF,则此椭圆的离心率为?](/uploads/image/z/2587891-67-1.jpg?t=%E5%B7%B2%E7%9F%A5%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%2CA%E4%B8%BA%E5%B7%A6%E9%A1%B6%E7%82%B9%2CB%E4%B8%BA%E7%9F%AD%E8%BD%B4%E4%B8%80%E9%A1%B6%E7%82%B9%2CF%E4%B8%BA%E5%8F%B3%E7%84%A6%E7%82%B9%2C%E4%B8%94%E5%90%91%E9%87%8FAB%E2%8A%A5%E5%90%91%E9%87%8FBF%C2%B7%C2%B7%C2%B7%E5%B7%B2%E7%9F%A5%E6%A4%AD%E5%9C%86x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2%3D1%28a%3Eb%3E0%29%2CA%E4%B8%BA%E5%B7%A6%E9%A1%B6%E7%82%B9%2CB%E4%B8%BA%E7%9F%AD%E8%BD%B4%E4%B8%80%E9%A1%B6%E7%82%B9%2CF%E4%B8%BA%E5%8F%B3%E7%84%A6%E7%82%B9%2C%E4%B8%94%E5%90%91%E9%87%8FAB%E2%8A%A5%E5%90%91%E9%87%8FBF%2C%E5%88%99%E6%AD%A4%E6%A4%AD%E5%9C%86%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA%3F)
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF···已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF,则此椭圆的离心率为?
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF···
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF,则此椭圆的离心率为?
已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF···已知椭圆x^2/a^2+y^2/b^2=1(a>b>0),A为左顶点,B为短轴一顶点,F为右焦点,且向量AB⊥向量BF,则此椭圆的离心率为?
|AB|^2=|BO|^2+|AO|^2=a^2+b^2
|BF|^2=|BO|^2+|OF|^2=b^2+c^2
|AF|^2=(|OF|+|OA|)^2=(c+a)^2
向量AB⊥向量BF,即
|AB|^2+|BF|^2=|AF|^2
a^2+b^2+b^2+c^2=(c+a)^2
a^2+2(a^2-c^2)+c^2=a^2+2ac+c^2
c^2+ac-a^2=0
e^2+e-1=0
e=(√5-1)/2或e=(-√5-1)/2(舍去)
所以e=(√5-1)/2
∵∠AOF=∠AOB=90(O为原点)
∠BAF=∠OBF
∴三角形AOB∽三角形BOF
∴OB/OA=BF/OB
所以b/a=c/b
∴b^2=ac
所以a^2-c^2=ac
∴(c/a)^2+c/a-1=0
所以离心率e=c/a=((根号5)-1)/2