椭圆x²/a²+y²/b²=1与直线x+y-1=0相交与AB两点当椭圆e范围为[√3/3,√2/2 ]且OA*OB=0时 求长轴长范围
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椭圆x²/a²+y²/b²=1与直线x+y-1=0相交与AB两点当椭圆e范围为[√3/3,√2/2 ]且OA*OB=0时 求长轴长范围
椭圆x²/a²+y²/b²=1与直线x+y-1=0相交与AB两点
当椭圆e范围为[√3/3,√2/2 ]且OA*OB=0时 求长轴长范围
椭圆x²/a²+y²/b²=1与直线x+y-1=0相交与AB两点当椭圆e范围为[√3/3,√2/2 ]且OA*OB=0时 求长轴长范围
设A(x1,y1),B(x2,y2).则x1,x2是x^2/a^2+y^2/b^2=1和X+Y=1联立方程的2根.
联立方程得:(a^2+b^2)x^2-2a^2x+a^2-a^2*b^2=0.
x1x2=(a^2-a^2*b^2)/(a^2+b^2),x1+x2=2a^2/(a^2+b^2).
OA垂直OB==>x1x2+y1y2=0,y=1-x.前式计算整理得(分子):a^2+b^2-2a^2*b^2=0
b^2=a^2/(2a^2-1).
又∵√3/3≤e≤√2/2
∴1/3≤e^2≤1/2
∴1/3≤(a^2-b^2)/a^2≤1/2
即1/3≤[a^2-a^2/(2a^2-1)]/a^2≤1/2
∴1/3≤1-1/(2a^2-1)≤1/2
∴1/2≤1/(2a^2-1)≤2/3
∴3/2≤2a^2-1≤2
∴5≤4a^2≤6
∴√5≤2a≤√6