椭圆x^2/a^2+y^2/b^2=1（a＞b＞0）与直线x+y-1=0相交于P,Q两点,且OP⊥OQ（O为原点）,求1/a^2+1/b^2

椭圆x^2/a^2+y^2/b^2=1（a＞b＞0）与直线x+y-1=0相交于P,Q两点,且OP⊥OQ（O为原点）,求1/a^2+1/b^2
椭圆x^2/a^2+y^2/b^2=1（a＞b＞0）与直线x+y-1=0相交于P,Q两点,且OP⊥OQ（O为原点）,求1/a^2+1/b^2

椭圆x^2/a^2+y^2/b^2=1（a＞b＞0）与直线x+y-1=0相交于P,Q两点,且OP⊥OQ（O为原点）,求1/a^2+1/b^2
设P（x1,y1）Q(x2,y2),x+y-1=0带入椭圆方程得（a²＋b²）x²-2a²x+a²-a²b²＝0,
x1+x2=2a²／(a²＋b²),x1x2=(a²-a²b²)／(a²＋b²).
y1y2=(1-x1)(1-x2)=1-(x1+x2)+x1x2=(b²-a²b²)／(a²＋b²).
OP垂直OQ得,x1x2+y1y2=0,即(a²-a²b²)／(a²＋b²)+(b²-a²b²)／(a²＋b²)=0.
整理得1/a^2+1/b^2=2.