Question

已知F1,F2是椭圆x^2/a^2+y^2/b^2=1(a>b>0)的两个焦点,O为坐标原点,点P(-1,根号2/2)在

Answer 1

PM+F2M=0

M中点

Mx=(Px+F2x)/2=0 F2x=-Px=1,c=1

x^2/a^2+y^2/(a^2-1)=1

1/a^2+(1/2)/(a^2-1)=1

a^2(a^2-1)=a^2-1+(1/2)a^2

a^4-(5/2)a^2+1=0

(a^2-2)(a^2-1/2)=0

a^2=2,b^2=2-1=1

方程x^2/2+y^2=1

圆O：x^2+y^2=1

y=kx+m AyBy=k^2 AxBx+km(Ax+Bx)+m^2

(1+k^2)x^2+2kmx+m^2-1=0

判别式

(2km)^2-4(1+k^2)(m^2-1)=0

k^2=m^2-1

x^2/2+y^2=1

(1/2+k^2)x^2+2kmx+(m^2-1)=0

Ax+Bx=-2km/(1/2+k^2)

AxBx=(m^2-1)/(1/2+k^2)

OA(Ax,Ay) OB(Bx,By)

OA*OB=AxBx+AyBy=(m^2-1)/(1/2+k^2)+k^2(m^2-1)/(1/2+k^2)+m^2+2km(-2km)/(1/2+k^2)

=[(m^2-1)+(m^2-1)^2+m^2(m^2-1/2)-4(m^2-1)m^2] / (m^2-1/2)

=3/2

2[m^2-1+m^4-2m^2+1+m^4-(1/2)m^2-4m^4+4m^2]=3m^2-3/2

2(-2m^4+(1/2)m^2+1)=3m^2-3/2

4m^4-4m^2=7/2

(2m^2-1)^2=9/2

2m^2-1=1+3√2/2

m^2=1+3√2/4

m^2-1=3√2/4

k^2=3√2/4

k=√(3√2/4)或k=-√(3√2/4)