已知P是椭圆x²/a²+y²/b²=1﹙a>b>0﹚一动点,且P与椭圆长轴两顶点连线的斜率之积为-1/2,求椭圆离心率
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![已知P是椭圆x²/a²+y²/b²=1﹙a>b>0﹚一动点,且P与椭圆长轴两顶点连线的斜率之积为-1/2,求椭圆离心率](/uploads/image/z/5265643-67-3.jpg?t=%E5%B7%B2%E7%9F%A5P%E6%98%AF%E6%A4%AD%E5%9C%86x%26%23178%3B%EF%BC%8Fa%26%23178%3B%2By%26%23178%3B%EF%BC%8Fb%26%23178%3B%3D1%EF%B9%99a%EF%BC%9Eb%EF%BC%9E0%EF%B9%9A%E4%B8%80%E5%8A%A8%E7%82%B9%2C%E4%B8%94P%E4%B8%8E%E6%A4%AD%E5%9C%86%E9%95%BF%E8%BD%B4%E4%B8%A4%E9%A1%B6%E7%82%B9%E8%BF%9E%E7%BA%BF%E7%9A%84%E6%96%9C%E7%8E%87%E4%B9%8B%E7%A7%AF%E4%B8%BA-1%2F2%2C%E6%B1%82%E6%A4%AD%E5%9C%86%E7%A6%BB%E5%BF%83%E7%8E%87)
已知P是椭圆x²/a²+y²/b²=1﹙a>b>0﹚一动点,且P与椭圆长轴两顶点连线的斜率之积为-1/2,求椭圆离心率
已知P是椭圆x²/a²+y²/b²=1﹙a>b>0﹚一动点,
且P与椭圆长轴两顶点连线的斜率之积为-1/2,求椭圆离心率
已知P是椭圆x²/a²+y²/b²=1﹙a>b>0﹚一动点,且P与椭圆长轴两顶点连线的斜率之积为-1/2,求椭圆离心率
且P与椭圆长轴两顶点连线的斜率之积为-1/2
设P(x,y),则x²/a²+y²/b²=1, 即y²=b²*(1-x²/a²)=(b²/a²)(a²-x²) ①
两个长轴顶点是A(-a,0),B(a,0)
∴ [y/(x+a)] *[y/(x-a)]=-1/2
y²=(-1/2)(x²-a²) ②
由①②
∴ (b²/a²)(a²-x²)=(-1/2)(x²-a²)
∴ b²/a²=1/2
即 a²=2b²=2(a²-c²)
∴ a²=2c²
∴ a=√2 c
∴ 离心率e=c/a=√2/2
√2/2