Question

高三数学 谢啦！

Answer 1

设l:y=2(x+a),A(-a,0),C(0,2a),B(x0,y0),
由AB=6/13BC得,
x0=(-a+6/13*0)/(19/13)=-13/19 a
y0=(0+6/13*2a)/(19/13)=12/19a,
(-13/19 a,12/19 a)在椭圆上，
(13/19)^2+(12/19)^2 a^2/b^2=1,
a/b=2/sqrt(3)
离心率c/a=1/2
2.
椭圆方程化为x^2/(4c^2)+y^2/(3c^2)=1,设P(x1,y1)
由y=kx+m,x^2/(4c^2)+y^2/(3c^2)=1得,
(4k^2+3)x^2+8kmx+4(m^2-3c^2)=0,
由已知
2x1=-8km/(4k^2+3),64k^2 m^2-16(4k^2+3)(m^2-3c^2)=0,
x1=-4km/(4k^2+3),
m^2=(4k^2+3)c^2,@1
Q(4,4k+m),M(1,0)
PM⊥QM
(x1-1,y1) .（3,4k+m)=0,
3(x1-1)+(4k+m)(kx1+m)=0,
(4k^2+km+3)x1+4km+m^2-3=0,
x1=-(4km+m^2-3)/(4k^2+km+3)

4km/(4k^2+3)=(4km+m^2-3)/(4k^2+km+3)
m^2=4k^2+3,@2
将@1代入@2得,c^2=1,c=1,
椭圆x^2/4+y^2/3=1