已知直线l过抛物线y=x²/4的焦点F和F关于直线x+y=0的对称点F',椭圆的中心在坐标原点o焦点在坐标轴上,直线l与椭圆交于P,Q

1
y=x^2/4,2p=4,p/2=1,焦点F(0,1)
F'(x',y'),FF'中点M(x'',y'')
x''=x'/2,y''=1+y'/2
直线l过FF',FF'A关于x+y=0对称,M在x+y=0上,x''=-y'' x'=-1-y'
l:y=kx+b 垂直于x+y=0,k=-1/-1=1
y=x+b 过F、F‘b=1,
直线l:y=x+1
y'=x'+1 x'=-1,y'=0,x''=-1/2,y''=1/2 F'(-1,0）,M(-1/2,1/2)
2
椭圆中心在原点,x^2/a^2+y^2/b^2=1
l:y=x+1相交P(X1,Y1),Q(X2,Y2)
(b^2+a^2)x^2+2a^2x+a^2-a^2b^2
X1+X2=-2a^2/(a^2+b^2)
X1X2=（a^2-a^2b^2)/(a^2+b^2)
Y1Y2=X1X2+X1+X2+1
OP:y=（Y1/X1)x OQ:y=(Y2/X2) OP垂直OQ,Y1/X1=-X2/Y2
Y1Y2+X1X2=0
2X1X2=-(X1+X2)-1
2(a^2-a^2b^2)=2a^2-(a^2+b^2)
2a^2b^2=a^2+b^2
(X1-X2)^2=4a^4/(a^2+b^2)^2 -4(a^2-a^2b^2)/(a^2+b^2)
(X1-X2)^2=1/b^4-4(a^2-a^2b^2)/(2a^2b^2)=1/b^4-2/b^2+2
PQ=(X1-X2)^2+(Y1-Y2)^2=2(X1-X2)^2,PQ=√10/2
(X1-X2)^2=5/4
1/b^4-2/b^2+2=5/4
(1/b^2-1)^2=1/4
1/b^2=1/2+1或1/b^2=1/2
b^2=2/3 或b^2=2
2a^2b^2=a^2+b^2
b^2=2/3,4a^2/3=a^2+2/3 a^2=2 椭圆方程x^2/2+y^2/(2/3)=1
b^2=2,4a^2=a^2+2 a^2=2/3 椭圆方程x^2/(2/3)+y^2/2=1