Question

详细求解过程，带图

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Answer 1

1
y=x+7/3
x^2/a^2+y^2/b^2=1
x^2/a^2+(x+7/3)^2/b^2=1
(1/a^2+1/b^2)x^2+14x/3b^2+49/9b^2-1=0
x1+x2=-14/3b^2/(1/a^2+1/b^2)=-a^214x/3/(b^2+a^2)=-a^214/3(2a^2-c^2)=-14/3(2-c^2/a^2)
(x1+x2)/2=-4/3 -7/3(2-e^2)=-4/3 7=8-4e^2 e^2=1/4
e=1/2

2
AB过O(0,0)
y=kx
x^2/a^2+y^2/b^2=1
(1+k^2)x^2/a^2=1
x1+x2=0
y1+y2=0
A(x1,y1)B(-x1,-y1)
M(x0,y0),
MA直线 y-y1=[(y1-y0)/(x1-x0)](x-x1) 斜率 (y1-y0)/(x1-x0)
MB直线 y+y1=[(-y1-y0)/(-x1-x0)](x+x1) 斜率 (y1+y0)/(x1+x0)
MAMB直线斜率积 [(y1-y0)(y1+y0)]/[(x1-x0)(x1+x0)]=(y1^2-y0^2)/(x1^2-x0^2)
x0^2/a^2+y0^2/b^2=1 1)
x1^2/a^2+y1^2/b^2=1 2)
2)-1) (x1^2-x0^2)/a^2=(y0^2-y1^2)/b^2
(y1^2-y0^2)/(x1^2-x0^2)=-b^2/a^2
斜率积为-b^2/a^2