3. Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ have its left and right foci at $F_{1}, F_{2}$, respectively. Point $A$ is an intersection point of the line passing through $F_{2}$ with a slope angle of $\frac{\pi}{4}$ and the hyperbola. If $\triangle F_{1} F_{2} A$ is an isosceles right triangle, then the eccentricity of the hyperbola is
A. $\frac{\sqrt{3}+1}{2}$
B. $\sqrt{3}+1$
C. $\frac{\sqrt{2}+1}{2}$
D. $\sqrt{2}+1$
According to the problem, $\frac{b^{2}}{a}=2 c \Rightarrow b^{2}=c^{2}-a^{2}=2 a c \Rightarrow e^{2}-2 e-1=0 \Rightarrow e=\sqrt{2}+1$, so the answer is $D$.(or: $e=\frac{2 c}{2 a}=\frac{1}{\sqrt{2}-1}=\sqrt{2}+1$.)