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 $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B }$ and $\overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$ , then the eccentricity of $C$ is $\_\_\_\_$ . Section III: Solution Questions: Total 70 points. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22 and 23 are optional; choose one to answer.
(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 $\overrightarrow { F _ { 1 } A } = \overrightarrow { A B }$ and $\overrightarrow { F _ { 1 } B } \cdot \overrightarrow { F _ { 2 } B } = 0$ , then the eccentricity of $C$ is $\_\_\_\_$ .
Section III: Solution Questions: Total 70 points. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22 and 23 are optional; choose one to answer.
\section*{(I) Required Questions: Total 60 points}