1. 设椭圆方程为x^2/a^2+y^2/b^2=1 右焦点(c,0) e=c/a=√2/2 a=√2c a=√2b
x=c代入 椭圆方程 c^2/2c^2+y^2/c^2=1 y=±c*√2/2
弦长=|y1-y2|=c*√2= 根号2 c=b=1 a=√2
椭圆方程为x^2/2+y^2=1
x^2+2y^2=2
2. 直线L y=kx+b P(x1,y1) Q(x2,y2)
OP⊥OQ x1*x2+y1*y2=0
联立x^2+2y^2=2和 y=kx+b
(1+2k^2)x^2+4kbx+2b^2-2=0 x1x2=(2b^2-2)/(1+2k^2) x1+x2=-4kb/(1+2k^2)
y1*y2=k^2x1x2+kb(x1+x2)+b^2
x1*x2+y1*y2=(1+k^2)(2b^2-2)/(1+2k^2)-4k^2b^2/(1+2k^2)+b^2=0
3b^2=2+2k^2
b^2=(2+2k^2)/3 |b|=√6*√(1+k^2)/3
点O到直线L的距离=|b|/√(1+k^2)=√6/3
O到直线L的距离是为定值=√6/3