已知曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2根号3/3,直线L过点(a,0),B(0,-b)两点,原点O到L的距离是已知双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2倍根3/3,过A(0,-b)与B(a,0)的直线与原点的距离为根号3/2.(1
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/27 15:30:29
![已知曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2根号3/3,直线L过点(a,0),B(0,-b)两点,原点O到L的距离是已知双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2倍根3/3,过A(0,-b)与B(a,0)的直线与原点的距离为根号3/2.(1](/uploads/image/z/3506344-16-4.jpg?t=%E5%B7%B2%E7%9F%A5%E6%9B%B2%E7%BA%BFx%5E2%2Fa%5E2-y%5E2%2Fb%5E2%3D1%28a%3E0%2Cb%3E0%29%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87e%3D2%E6%A0%B9%E5%8F%B73%2F3%2C%E7%9B%B4%E7%BA%BFL%E8%BF%87%E7%82%B9%28a%2C0%29%2CB%280%2C-b%29%E4%B8%A4%E7%82%B9%2C%E5%8E%9F%E7%82%B9O%E5%88%B0L%E7%9A%84%E8%B7%9D%E7%A6%BB%E6%98%AF%E5%B7%B2%E7%9F%A5%E5%8F%8C%E6%9B%B2%E7%BA%BFx%5E2%2Fa%5E2-y%5E2%2Fb%5E2%3D1%EF%BC%88a%3E0%2Cb%3E0%EF%BC%89%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87e%3D2%E5%80%8D%E6%A0%B93%2F3%2C%E8%BF%87A%280%2C-b%29%E4%B8%8EB%28a%2C0%29%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%B8%8E%E5%8E%9F%E7%82%B9%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BA%E6%A0%B9%E5%8F%B73%2F2.%281)
已知曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2根号3/3,直线L过点(a,0),B(0,-b)两点,原点O到L的距离是已知双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2倍根3/3,过A(0,-b)与B(a,0)的直线与原点的距离为根号3/2.(1
已知曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2根号3/3,直线L过点(a,0),B(0,-b)两点,原点O到L的距离是
已知双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2倍根3/3,过A(0,-b)与B(a,0)的直线与原点的距离为根号3/2.
(1)求此双曲线方程
(2)过点B作直线m交双曲线于M、N两点,若OM*ON=-23,求直线m的方程.
已知曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2根号3/3,直线L过点(a,0),B(0,-b)两点,原点O到L的距离是已知双曲线x^2/a^2-y^2/b^2=1(a>0,b>0)的离心率e=2倍根3/3,过A(0,-b)与B(a,0)的直线与原点的距离为根号3/2.(1
(1)离心率e=c/a=2√3/3 => c²/a²=4/3 =>b²/a²=c²/a²-1=1/3 ①
直线l斜率为b/a,方程为y=bx/a -b 即bx-ay-ab=0
原点到直线l的距离√3/2=|0+0-ab|/√(a²+b²) ②
①②两式联立解得:a=√3,b=1,c=2
双曲线方程为x²/3 - y² =1
(2)设直线m方程为y=kx-1,与双曲线方程联立得:(3k²-1)x²-6kx+6=0
由韦达定理可知:x1+x2=6k/(3k²-1),x1x2=6/(3k²-1)
y1y2=(kx1-1)(kx2-1)=k²x1x2-k(x1+x2)+1=6k²/(3k²-1)-6k²/(3k²-1) +1=1
(向量OM)·(向量ON)= -23=x1x2+y1y2=6/(3k²-1) +1
解得k=±1/2
所以直线m方程为y=1/2x-1或y=-1/2x-1