椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√2/2,直线x+y+1=0交于椭圆P.Q两点,且OP垂直OQ,求椭圆方程

椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√2/2,直线x+y+1=0交于椭圆P.Q两点,且OP垂直OQ,求椭圆方程
椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√2/2,直线x+y+1=0交于椭圆P.Q两点,且OP垂直OQ,求椭圆方程

椭圆x^2/a^2+y^2/b^2=1(a>b>0)的离心率e=√2/2,直线x+y+1=0交于椭圆P.Q两点,且OP垂直OQ,求椭圆方程
e²=1/2=c²/a²
a²=2c²
则b²=a²-c²=c²
所以是x²+2y²=2c²
y=-x-1
3x²+4x+2-2c²=0
x1+x2=-4/3
x1x2=(2-2c²)/3
y1y2=(-x1-1)(-x2-1)=x1x2+(x1+x2)+1=(1-2c²)/3
垂直则(y1/x1)(y2/x2)=-1
所以y1y2+x1x2=0
(2-2c²)/3+(1-2c²)/3=0
c²=3/4
a²=3/2
所以2x²/3+4y²/3=1

e=√2/2,a^2=2b^2=2c^2 P(x1,y1) Q(x2,y2) x^2+2y^2=2b^2 x=-1-y 代入得
3y^2-2y+(1-2b^2)=0 y1y2=(1-2b^2)/3 y1+y2=2/3
OP垂直OQ x1x2+y1y2=0 x1x2=y1y2+(y1+y2) +1
2y1y2+(y1+y2) +1=2(1-2b^2)/3+2/3+1=0 b^2=7/4 a^2=7/2
椭圆方程x^2/(7/2)+y^2/(7/4)=1