求过点e(0,-4)的直线l使其交椭圆于点m,n,且满足om·on=16/7

Question

求过点e(0,-4)的直线l使其交椭圆于点m,n,且满足om·on=16/7

Answer 1

1）设椭圆P的方程是：x^2/a^2+y^2/b^2=1
代入点A得：b^2=12,c/a=1/2==>c^2/a^2=1/4==>(a^2-b^2)/a^2=1/4==>a^2=16
即：椭圆方程是：x^2/16+y^2/12=1
(2)设：R(x1,y1),T(x2,y2)
向量OR*向量OT=(x1,y1)*(x2,y2)=x1x2+y1y2=16/7
直线L:y=kx-4代入椭圆方程
得：x^2/16+(kx-4)^2/12=1 ==>x^2(3+4k^2)-32kx+16=0
==>x1x2=16/(3+4k^2),
==>x1+x2=32k/(3+4k^2) y1*y2=(kx1-4)(kx2-4)=(k^2(x1x2)-4k(x1+x2)+16
所以：x1x2+y1y2=(1+k^2)x1x2-4k(x1+x2)+16=16/7
==>k^2=1 ==>k=±1
即：存在过点E(0,-4)的直线L：y=±x-4