e=c/a=√(a^2-b^2)/a=√6/3
(a^2-b^2)/a^2=2/3
a^2=3b^2 (1)
直线AB方程为:
bx-ay-ab=0
ab/√(a^2+b^2)=√3/2
a^2b^2/(a^2+b^2)=3/4 (2)
联立(1)(2)解得:
a^2=3 a=√3
b^2=1 b=1
椭圆方程为:
x^2/3+y^2=1 (3)
y=kx+2 (4)
(4)代入(3)得
x^2/3+(kx+2)^2=1
(3k^2+1)x^2+12kx+9=0
x1+x2=-12k/(3k^2+1)
x1x2=9/(3k^2+1)
y1+y2=kx1+2+kx2+2=k(x1+x2)+4=8/(3k^2+1)
y1y2=(kx1+2)(kx2+2)=k^2x1x2+2k(x1+x2)+4=(-11k^2-5)/(3k^2+1)
设圆心坐标为x0,y0
x0=(x1+x2)/2=-6/(3k^2+1) y0=-6k/(3k^2+1)+2=2/(3k^2+1)
如圆过E点,则有:
(x2-x1)^2+(y2-y1)^2=(x0+1)^2+y0^2
(x2+x1)^2-4x1x2+(y1+y2)^2-4y1y2=(3k^2-5)^2/(3k^2+1)^2+2/(3k^2+1)
解得
k=1.06127
存在K的值.使以CD为直径的圆过E点