椭圆定值在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,P与长轴两端点A,B的连线分别交短轴所在直线于M,N两点,设O为原点,求证:/OM/乘/ON/为定值!在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,椭圆短轴上的两个
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/30 02:22:44
![椭圆定值在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,P与长轴两端点A,B的连线分别交短轴所在直线于M,N两点,设O为原点,求证:/OM/乘/ON/为定值!在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,椭圆短轴上的两个](/uploads/image/z/12963315-3-5.jpg?t=%E6%A4%AD%E5%9C%86%E5%AE%9A%E5%80%BC%E5%9C%A8%E6%A4%AD%E5%9C%86X2%2Fa2+%2BY2%2Fb2%3D1%EF%BC%88a%3Eb%3E0%EF%BC%89%E4%B8%8A%E5%8F%96%E4%B8%80%E7%82%B9P%2CP%E4%B8%8E%E9%95%BF%E8%BD%B4%E4%B8%A4%E7%AB%AF%E7%82%B9A%2CB%E7%9A%84%E8%BF%9E%E7%BA%BF%E5%88%86%E5%88%AB%E4%BA%A4%E7%9F%AD%E8%BD%B4%E6%89%80%E5%9C%A8%E7%9B%B4%E7%BA%BF%E4%BA%8EM%2CN%E4%B8%A4%E7%82%B9%2C%E8%AE%BEO%E4%B8%BA%E5%8E%9F%E7%82%B9%2C%E6%B1%82%E8%AF%81%EF%BC%9A%2FOM%2F%E4%B9%98%2FON%2F%E4%B8%BA%E5%AE%9A%E5%80%BC%21%E5%9C%A8%E6%A4%AD%E5%9C%86X2%2Fa2+%2BY2%2Fb2%3D1%EF%BC%88a%3Eb%3E0%EF%BC%89%E4%B8%8A%E5%8F%96%E4%B8%80%E7%82%B9P%EF%BC%8C%E6%A4%AD%E5%9C%86%E7%9F%AD%E8%BD%B4%E4%B8%8A%E7%9A%84%E4%B8%A4%E4%B8%AA)
椭圆定值在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,P与长轴两端点A,B的连线分别交短轴所在直线于M,N两点,设O为原点,求证:/OM/乘/ON/为定值!在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,椭圆短轴上的两个
椭圆定值
在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,P与长轴两端点A,B的连线分别交短轴所在直线于M,N两点,设O为原点,求证:/OM/乘/ON/为定值!
在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,椭圆短轴上的两个端点分别是B1 B2 若直线PB1,PB2分别与X轴交于M,N,设O为原点,求证:/OM/乘/ON/为定值!
椭圆定值在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,P与长轴两端点A,B的连线分别交短轴所在直线于M,N两点,设O为原点,求证:/OM/乘/ON/为定值!在椭圆X2/a2 +Y2/b2=1(a>b>0)上取一点P,椭圆短轴上的两个
x^2/a^2+y^2/b^2=1,(a>b>0)
设P点坐标为(m,n)
长轴两端点坐标为A(a,0)B(-a,0)则
m^2/a^2+n^2/b^2=1 即(a^2n^2)/(a^2-m^2)=b^2
PA的直线方程为
y-0=[(n-0)/(m-a)](x-a)
令x=0,则y=-an/(m-a),即
M点坐标为(-an/(m-a),0)
PB的直线方程为
y-0=[(n-0)/(m+a)](x+a)
令x=0,则x=an/(m+a),即
N点坐标为(an/(m+a),0)
|OM|*|ON|=|-an/(m-a)|*|an/(m+a)|=|(a^2n^2)/(a^2-m^2)|=b^2
所以|OM|*|ON|为定值b^2
x^2/a^2+y^2/b^2=1,(a>b>0)
设P点坐标为(m,n)
短轴两端点坐标为B1(0,b)B2(0,-b)则
m^2/a^2+n^2/b^2=1 即(b^2m^2)/(a^2-n^2)=a^2
PB1的直线方程为
y-b=[(n-b)/(m-0)](x-0)
令y=0,则x=-bm/(n-b),即
M点坐标为(-bm/(n-b),0)
PB2的直线方程为
y+b=[(n+b)/(m-0)](x-0)
令y=0,则x=bm/(n+b),即
N点坐标为(bm/(n+b),0)
|OM|*|ON|=|-bm/(n-b)|*|bm/(n+b)|=|(b^2m^2)/(a^2-n^2)|=a^2
所以|OM|*|ON|为定值a^2