1、e=c/a=1/2,c=a/2,b^2=a^2-c^2=3a^2/4,
b=√3a/2,
设椭圆方程为:x^2/a^2-y^2/(3a^2/4)=1,
圆的方程为:x^2+y^2=3a^2/4,
直线y=x+√6,代入圆方程,
x^2+(x+√6)^2-3a^2/4=0,
2x^2+2√6x+6-3a^2/4=0,
当直线和圆相切时,判别式b^2-4ac=0,
a=2,
b=3*/4=3,c=1,
故方程为:x^2/4+y^2/3=1.
2.设A(m,n), B(m,-n), BP: x=ky+4,代入椭圆方程,
得(3k^2+4)y^2+24ky+36=0
y1=-n, y2=y1y2/y1=36/[-n(3k^2+4)],
结合k=(4-m)/n,
得y2=36/[-n( (3(m-4)^2+4n^2)/n^2]
又m^2/4 + n^2/3=1,
得3m^2+4n^2=12,
故y2=3n/(2m-5)
E(x2,y2), x2=ky2+4=(4-m)/n*3n/(2m-5)+4=(5m-8)/(2m-5)
AE: (y-n)/(x-m)=(y2-n)/(x2-m)=n(m-4)/[(m-1)(m-4)]=n/(m-1)
令Y=0, 得x=1,
即恒过Q(1,0)
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