点P为椭圆x²/a²+y²/b²=1(a>b>0)上任意一点,异于顶点,椭圆短轴的两个端点分别是B1,B2,若直线PB1,PB2分别与x轴交于点M,N求证:OM*ON为定值
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/29 22:04:02
![点P为椭圆x²/a²+y²/b²=1(a>b>0)上任意一点,异于顶点,椭圆短轴的两个端点分别是B1,B2,若直线PB1,PB2分别与x轴交于点M,N求证:OM*ON为定值](/uploads/image/z/5498852-68-2.jpg?t=%E7%82%B9P%E4%B8%BA%E6%A4%AD%E5%9C%86x%26%23178%3B%2Fa%26%23178%3B%2By%26%23178%3B%2Fb%26%23178%3B%3D1%EF%BC%88a%3Eb%3E0%EF%BC%89%E4%B8%8A%E4%BB%BB%E6%84%8F%E4%B8%80%E7%82%B9%2C%E5%BC%82%E4%BA%8E%E9%A1%B6%E7%82%B9%2C%E6%A4%AD%E5%9C%86%E7%9F%AD%E8%BD%B4%E7%9A%84%E4%B8%A4%E4%B8%AA%E7%AB%AF%E7%82%B9%E5%88%86%E5%88%AB%E6%98%AFB1%2CB2%2C%E8%8B%A5%E7%9B%B4%E7%BA%BFPB1%2CPB2%E5%88%86%E5%88%AB%E4%B8%8Ex%E8%BD%B4%E4%BA%A4%E4%BA%8E%E7%82%B9M%2CN%E6%B1%82%E8%AF%81%EF%BC%9AOM%2AON%E4%B8%BA%E5%AE%9A%E5%80%BC)
点P为椭圆x²/a²+y²/b²=1(a>b>0)上任意一点,异于顶点,椭圆短轴的两个端点分别是B1,B2,若直线PB1,PB2分别与x轴交于点M,N求证:OM*ON为定值
点P为椭圆x²/a²+y²/b²=1(a>b>0)上任意一点,异于顶点,椭圆短轴的两个端点分别是B1,B2,若直线PB1,PB2分别与x轴交于点M,N求证:OM*ON为定值
点P为椭圆x²/a²+y²/b²=1(a>b>0)上任意一点,异于顶点,椭圆短轴的两个端点分别是B1,B2,若直线PB1,PB2分别与x轴交于点M,N求证:OM*ON为定值
证明:
设 p点为(x0,y0)
则:x0²/a²+y0²/b²=1,整理后为:b²x0²+a²y0²=a²b² ,
后面会用到它的变形 -b²x0²=a²y0²-a²b² (1)
直线B1M:(y0-b)/x0=y-b/x,与x轴交点M的横坐标为:-bx0/(y0-b) /即y=0,解出的x/
同理N坐标为:bx0/(y0+b)
OM*ON=-b²x0²/(y0²-b²) 带入(1)得:OM*ON=a².