It is known that F1F2 is the left and right focus of hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 (a > 0, b > 0), If the triangle abf2 is an acute triangle, then the value range of eccentricity of the hyperbola

It is known that F1F2 is the left and right focus of hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 (a > 0, b > 0), If the triangle abf2 is an acute triangle, then the value range of eccentricity of the hyperbola

∵ F1 is the left focus
∴F1A>F2A
It must be an acute angle
∵ ab ⊥ X axis
∴F2A=F2B
∠F1AF2=∠F1BF2
∵ triangle abf2 is an acute triangle
Only ∠ af2b is an acute angle
∵∠AF2F1=∠BF2F1=1/2∠AF2B<1/2*90°=45°
∴∠AF2F1=∠BF2F1<45°
Substituting x = - C into x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1
c^2/a^2-y^2/b^2=1
y=±b^2/a
∴AF1=b^2/a
F1F2=2c
tan∠AF2F1=AF1/F1F2<1
b^2/(2ac)<1
c^2-2ac-a^2<0
e^2-2e-1<0
1-√2<e<1+√2
∵ hyperbola
∴1<e<1+√2