椭圆C:x^2/a^2+y^2=1(a〉1),右焦点F,过F做倾斜角为45°的直线交椭圆于AB两点,若向量AF=2FB,求椭圆方程
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![椭圆C:x^2/a^2+y^2=1(a〉1),右焦点F,过F做倾斜角为45°的直线交椭圆于AB两点,若向量AF=2FB,求椭圆方程](/uploads/image/z/8566628-68-8.jpg?t=%E6%A4%AD%E5%9C%86C%EF%BC%9Ax%5E2%2Fa%5E2%2By%5E2%3D1%EF%BC%88a%E3%80%891%EF%BC%89%2C%E5%8F%B3%E7%84%A6%E7%82%B9F%2C%E8%BF%87F%E5%81%9A%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA45%C2%B0%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%BA%A4%E6%A4%AD%E5%9C%86%E4%BA%8EAB%E4%B8%A4%E7%82%B9%2C%E8%8B%A5%E5%90%91%E9%87%8FAF%3D2FB%2C%E6%B1%82%E6%A4%AD%E5%9C%86%E6%96%B9%E7%A8%8B)
椭圆C:x^2/a^2+y^2=1(a〉1),右焦点F,过F做倾斜角为45°的直线交椭圆于AB两点,若向量AF=2FB,求椭圆方程
椭圆C:x^2/a^2+y^2=1(a〉1),右焦点F,过F做倾斜角为45°的直线交椭圆于AB两点,
若向量AF=2FB,求椭圆方程
椭圆C:x^2/a^2+y^2=1(a〉1),右焦点F,过F做倾斜角为45°的直线交椭圆于AB两点,若向量AF=2FB,求椭圆方程
设直线AB交椭圆的右准线于点G,
作点A、B、F到椭圆的右准线的垂线,垂足分别为C、D、E;
则有:GC/GD = AC/BD = AF/BF = 2 ,CE/DE = AF/BF = 2 ,
可得:CE = 2DE ,GD = CD = CE+DE = 3DE ,GC = 2GD = 6DE ,GE = GD+DE = 4DE ,
所以,BD/EF = GD/GE = 3/4 ;
设右焦点F的坐标为 (c,0) ,
则有:a² = c²+1 ,椭圆的右准线为 x = a²/c = (c²+1)/c = c+1/c ;
所以,EF = c+1/c-c = 1/c ,BD = (3/4)EF = 3/(4c) ,
可得:点B的横坐标为 c+1/c-3/(4c) = c+1/(4c) ;
依题意,△GBD、△GFE 都是等腰直角三角形,
所以,DE = GE-GD = EF-BD = 1/(4c) ,
可得:点B的纵坐标为 1/(4c) ;
将点B的坐标代入椭圆方程,可得:[c+1/(4c)]²/a²+[1/(4c)]² = 1 ,
将 a² = c²+1 代入可得:[c+1/(4c)]²/(c²+1)+[1/(4c)]² = 1 ,
解得:c² = 2/7 ,
可得:a² = c²+1 = 9/7 ;
所以,椭圆方程为 7x²/9+y² = 1 .
a²=9/7