已知双曲线x2/a2-y2/b2=1(a>0,b>0)和x2/a2-y2/b2=-1的离心率分别为e1和e2

x^2/a^2-y^2/b^2=1和y^2/b^2-x^2/a^2=1互为共扼双曲线,离心率e1=c/a,e2=c/b,∴ 1/(e1)^2+1/(e2)^2=(a/c)^2+(b/c)^2=(a^2+b^2)/(c^2)=c^2/c^2=1.
(2) 由1=1/(e1)^2+1/(e2)^2=[(e1)^2+(e2)^2]/(e1·e2)^2≥2(e1·e2)/(e1·e2)^2=2/(e1·e2),∴ e1·e2≥2.
∴ e1+e2≥2√(e1·e2)≥2√2,当且仅当e1=e2,即a=b时,取