16. Given hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 1 }$ intersects the two asymptotes of $C$ at points $A , B$ respectively. If $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B } , \overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$, then the eccentricity of $C$ is $\_\_\_\_$. III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17-21 are required for all students. Questions 22 and 23 are optional; students should choose one to answer. If more than one is answered, only the first one will be graded. (I) Required Questions: Total 60 points.
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16. Given hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 1 }$ intersects the two asymptotes of $C$ at points $A , B$ respectively. If $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B } , \overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$, then the eccentricity of $C$ is $\_\_\_\_$.
III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17-21 are required for all students. Questions 22 and 23 are optional; students should choose one to answer. If more than one is answered, only the first one will be graded.
(I) Required Questions: Total 60 points.