已知双曲线C:x^2/a^2-y^2/b^2=1(a>0,b>0),的离心率为2,焦点到渐近线的距离为2倍根号3.点P的坐标为(0,-2),过P的直线L与双曲线C交于不同两点M,N(1)求双曲线C的方程;(2)设t=向量OM*向量OP+向量OM*向
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![已知双曲线C:x^2/a^2-y^2/b^2=1(a>0,b>0),的离心率为2,焦点到渐近线的距离为2倍根号3.点P的坐标为(0,-2),过P的直线L与双曲线C交于不同两点M,N(1)求双曲线C的方程;(2)设t=向量OM*向量OP+向量OM*向](/uploads/image/z/1611463-31-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%8F%8C%E6%9B%B2%E7%BA%BFC%3Ax%5E2%2Fa%5E2-y%5E2%2Fb%5E2%3D1%28a%3E0%2Cb%3E0%29%2C%E7%9A%84%E7%A6%BB%E5%BF%83%E7%8E%87%E4%B8%BA2%2C%E7%84%A6%E7%82%B9%E5%88%B0%E6%B8%90%E8%BF%91%E7%BA%BF%E7%9A%84%E8%B7%9D%E7%A6%BB%E4%B8%BA2%E5%80%8D%E6%A0%B9%E5%8F%B73.%E7%82%B9P%E7%9A%84%E5%9D%90%E6%A0%87%E4%B8%BA%280%2C-2%29%2C%E8%BF%87P%E7%9A%84%E7%9B%B4%E7%BA%BFL%E4%B8%8E%E5%8F%8C%E6%9B%B2%E7%BA%BFC%E4%BA%A4%E4%BA%8E%E4%B8%8D%E5%90%8C%E4%B8%A4%E7%82%B9M%2CN%EF%BC%881%EF%BC%89%E6%B1%82%E5%8F%8C%E6%9B%B2%E7%BA%BFC%E7%9A%84%E6%96%B9%E7%A8%8B%EF%BC%9B%EF%BC%882%EF%BC%89%E8%AE%BEt%3D%E5%90%91%E9%87%8FOM%2A%E5%90%91%E9%87%8FOP%2B%E5%90%91%E9%87%8FOM%2A%E5%90%91)
已知双曲线C:x^2/a^2-y^2/b^2=1(a>0,b>0),的离心率为2,焦点到渐近线的距离为2倍根号3.点P的坐标为(0,-2),过P的直线L与双曲线C交于不同两点M,N(1)求双曲线C的方程;(2)设t=向量OM*向量OP+向量OM*向
已知双曲线C:x^2/a^2-y^2/b^2=1(a>0,b>0),的离心率为2,焦点到渐近线的距离为2倍根号3.点P的坐标为(0,-2),过P的直线L与双曲线C交于不同两点M,N(1)求双曲线C的方程;(2)设t=向量OM*向量OP+向量OM*向量PN,求t的取值范围
主要是第二问.
已知双曲线C:x^2/a^2-y^2/b^2=1(a>0,b>0),的离心率为2,焦点到渐近线的距离为2倍根号3.点P的坐标为(0,-2),过P的直线L与双曲线C交于不同两点M,N(1)求双曲线C的方程;(2)设t=向量OM*向量OP+向量OM*向
(1)c=2a,b=2√3,12=3a^2,a^2=4,b^2=12,双曲线C的方程x^2/4 - y^2/12=1
(2)设M(x1,y1),N(x2,y2),则t=(OM)*(OP)+ (OM )*(PN)=-2y+x1x2+y1(y2+2)
1)易知焦点到渐近线的间隔为b=2√3,又e=c/a=2,易求a=4,故双曲线方程为x2/16-y2/12=1 2)记功点P的直线方程为y=kx-2,点M(x1,y1),N(x2,y2) 直线方程代入双曲线方程化简为:(3-4k2)x2+16kx-64=0 ☆ 则x1+x2=16k/(4k2-3),x1x2=64/(4k2-3), 又向量OM=(x1,y1),OP=(0,-2),PN=(x2,y2+2), 则t=(OM)* (OP)+ (OM )*(PN)=-2y1+x1x2+y1(y2+2)=(k2+1)x1x2-2k(x1+x2)+4 = (48k2+52)/(4k2-3)=12+88/(4k2-3) 又方程☆有两个不等实根,故有△=256k2+256(3-4k2)>0,且3-4k2≠0,得0≤k2<1且k2≠3/4 所以t≤-52/3或t>100